Linear and Abstract AlgebraLinear Systems of Equations, Eigenvalues and Eigenvectors, Vector Spaces, Groups, Rings, FieldsMon, 25 May 2020 20:16:03 +0000Mon, 25 May 2020 20:16:03 +0000Math Help Boards - Free Math Help
https://mathhelpboards.com/
Can I turn this equation into a linearly dependent square matrix under certain conditions?Sat, 23 May 2020 22:36:31 +0000
https://mathhelpboards.com/threads/can-i-turn-this-equation-into-a-linearly-dependent-square-matrix-under-certain-conditions.27577/
https://mathhelpboards.com/threads/can-i-turn-this-equation-into-a-linearly-dependent-square-matrix-under-certain-conditions.27577/invalid@example.com (Barbudania)BarbudaniaI have an equation that comes from an especific topic of cam mechanisms and it goes like this:
For this it doesn't matter what each variable means.
I'm trying to create a 3x3 matrix with a determinant equal to zero. The determinant is linearly dependent because I will create a nomogram from it and its vectors have to be on the same plane.
I did come up with this answer:
$$
det \begin{bmatrix} tan(B) &...
Can I turn this equation into a linearly dependent square matrix under certain conditions?]]>223-24 pivots and consistencySat, 23 May 2020 02:47:25 +0000
https://mathhelpboards.com/threads/23-24-pivots-and-consistency.27581/
https://mathhelpboards.com/threads/23-24-pivots-and-consistency.27581/invalid@example.com (karush)karush
ok a pivot column is one of $[0,,,0,b]$ where b is non zero.... b is 1 in RREF
not sure of the best answer but #23 there will be no free variables since 4 equation can derive fout answers altho some asnwera man be the same
#24 we can not derive a 4th or 5th answer with 3 equations]]>0complete augmented by row operationsFri, 22 May 2020 14:11:23 +0000
https://mathhelpboards.com/threads/complete-augmented-by-row-operations.27574/
https://mathhelpboards.com/threads/complete-augmented-by-row-operations.27574/invalid@example.com (karush)karush
$\left[
\begin{array}{rrrr|r}
1& -5& 4& 0&0\\
0& 1& 0& 1&0\\
0& 0& 3& 0&0\\
0& 0& 0& 2&0
\end{array}\right] $
OK my first move on this is $r_3/3$ and $r_4/2$.
]]>1LA 1.1.6 augmented MatrixFri, 22 May 2020 02:43:26 +0000
https://mathhelpboards.com/threads/la-1-1-6-augmented-matrix.27551/
https://mathhelpboards.com/threads/la-1-1-6-augmented-matrix.27551/invalid@example.com (karush)karushcomplete
$$\left[
\begin{array}{rrrr|r}
1& -6& 4& 0&-1\\
0& 2& -7& 0&4\\
0& 0& 1& 2&-3\\
0& 0& 4& 1&2\
\end{array}\right]$$
ok assume next step is $r_2/2$ and $r_4/4$ introducing fractions]]>7Practical issue regarding QuaternionsWed, 20 May 2020 12:06:08 +0000
https://mathhelpboards.com/threads/practical-issue-regarding-quaternions.27552/
https://mathhelpboards.com/threads/practical-issue-regarding-quaternions.27552/invalid@example.com (jensru)jensruHi there!
I´ve got a practical problem with quaternions which I was not able to solve by my own so far.
A machine detects the position and orientation of an object which I get as unity quaternion.
I visualize that using matlab, which still works more or less, but I´d like to 'force' the object to a defined position in the beginning and make the following movement relative to that. I hope it´s understandable what I want?
So my thought was to 'define' the first quaternion, which is ~ [0.89...
Practical issue regarding Quaternions]]>0Meaning of operation gxFri, 15 May 2020 15:29:02 +0000
https://mathhelpboards.com/threads/meaning-of-operation-gx.27449/
https://mathhelpboards.com/threads/meaning-of-operation-gx.27449/invalid@example.com (mathmari)mathmariHey!!
Let $G$ be a group and let $g\in G$. We define: \begin{align*}&\lambda_g:G\rightarrow G, \ x\mapsto gx \\ &\gamma_g:G\rightarrow G, \ x\mapsto gxg^{-1}\end{align*}
Show for all $g,h\in G$:
$\lambda_{1_G}=\text{id}_G$ and $\lambda_{gh}=\lambda_g\circ \lambda_h$
$\lambda_g$ is a permutation and it holds that $\lambda_g^{-1}=\lambda_{g^{-1}}$
The map $\lambda:G\rightarrow \text{Sym}(G), \ g\mapsto \lambda_g$ is a group monomorphism...
Meaning of operation gx]]>19Orthonormal basis - Set of all isometriesFri, 15 May 2020 14:23:40 +0000
https://mathhelpboards.com/threads/orthonormal-basis-set-of-all-isometries.27447/
https://mathhelpboards.com/threads/orthonormal-basis-set-of-all-isometries.27447/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n\in \mathbb{N}$ and $\mathbb{R}^n$. A basis $B=(b_1, \ldots, b_n)$ of $V$ is an orthonormal basis, if $b_i\cdot b_j=\delta_{ij}$ for all $1\leq i,j,\leq n$.
Let $E=(e_1, \ldots,e_n)$ be the standard basis and let $\phi \in O(V)$. ($O(V)$ is the set of all isometries $\alpha$ such that $\alpha(0_V)=0_V$)
I want to show the following:
Let $b_i:=\phi (e_i)$ for all $1\leq i\leq n$. Then $(b_1, \ldots , b_n)$ is an orthonormal basis.
For $v\in V$ it...
Orthonormal basis - Set of all isometries]]>23Group mono-, endo-, iso-, homomorphismFri, 15 May 2020 08:13:52 +0000
https://mathhelpboards.com/threads/group-mono-endo-iso-homomorphism.27455/
https://mathhelpboards.com/threads/group-mono-endo-iso-homomorphism.27455/invalid@example.com (mathmari)mathmariHey!!
Let $(G, \#), \ (H, \square )$ be groups. Show:
For $(g,h), (g',h')\in G\times H$ we define the operation $\star$ on $G\star H$ as follows:
\begin{equation*}\star: (G\times H)\times (G\times H,\star), \ \left ((g,h), (g',h')\right )\mapsto (g\# g', h\square h')\end{equation*} Then $(G\times H, \star)$ is a group.
The map $G\rightarrow G\times H, \ g\mapsto (g, e_H)$ is a monomorphism, where $e_H$ is the neutral element in $H$.
The map $G\times H\rightarrow...
Group mono-, endo-, iso-, homomorphism]]>9Ideals of a ring of matrixesThu, 14 May 2020 07:18:39 +0000
https://mathhelpboards.com/threads/ideals-of-a-ring-of-matrixes.27461/
https://mathhelpboards.com/threads/ideals-of-a-ring-of-matrixes.27461/invalid@example.com (Alex224)Alex224Hi.
I have this ring of matrixes: R = { \[ \begin{pmatrix} a & 0\\ b & c\\ \end{pmatrix} \]}
while a,b,c is from some field F.
now, I need to find all the ideals of this ring. I found five ideals. here there are:
now, Im kind of stuck to explain why there cannot be a six's ideal. I...
Ideals of a ring of matrixes]]>0Vector of polynomial and basis.Tue, 12 May 2020 12:33:49 +0000
https://mathhelpboards.com/threads/vector-of-polynomial-and-basis.27336/
https://mathhelpboards.com/threads/vector-of-polynomial-and-basis.27336/invalid@example.com (Displayer1243)Displayer1243If this question is in the wrong forum please let me know where to go.
For p
, the vector space of polynomials to the form ax'2+bx+c. p(x), q(x)=p(-1) 1(-1)+p(0), q(0)+p(1) q(1), Assume that this is an inner product. Let W be the subspace spanned by
.
a)...
Vector of polynomial and basis.]]>2Is the composition of the isometries a rotation?Sat, 09 May 2020 16:30:35 +0000
https://mathhelpboards.com/threads/is-the-composition-of-the-isometries-a-rotation.27368/
https://mathhelpboards.com/threads/is-the-composition-of-the-isometries-a-rotation.27368/invalid@example.com (mathmari)mathmariHey!!
Let $\delta_a:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the rotation around the origin with angle $\alpha$ and let $\sigma_{\alpha}:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the reflection about a line through the origin that has angle $\frac{\alpha}{2}$ with the $x$-axis.
Let $v\in V$ and $\alpha\in \mathbb{R}$.
I want to determine the geometric description of the isometries $\tau_v\circ \delta_{\alpha}\circ \tau_v^{-1}$ and $\tau_v\circ \sigma_{\alpha}\circ \tau_v^{-1}$...
Is the composition of the isometries a rotation?]]>7Subsets of permutation group: PropertiesSat, 09 May 2020 11:35:35 +0000
https://mathhelpboards.com/threads/subsets-of-permutation-group-properties.27359/
https://mathhelpboards.com/threads/subsets-of-permutation-group-properties.27359/invalid@example.com (mathmari)mathmariHey!!
Let $G$ be a permutation group of a set $X\neq \emptyset$ and let $x,y\in X$. We define:
\begin{align*}&G_x:=\{g\in G\mid g(x)=x\} \\ &G_{x\rightarrow y}:=\{g\in G\mid g(x)=y\} \\ &B:=\{y\in X\mid \exists g\in G: g(x)=y\}\end{align*}
Show the following:
$G_x$ is a subgroup of $G$.
The set $\{G_{x\rightarrow y}\mid y\in B\}\subseteq 2^G$ is a partition of $G$.
Let $g\in G_{x\rightarrow y}$ and let $u\in G_x$ and $v\in G_{x\rightarrow y}$. Then it holds that...
Subsets of permutation group: Properties]]>18Properties about vectorsSat, 09 May 2020 07:43:12 +0000
https://mathhelpboards.com/threads/properties-about-vectors.27360/
https://mathhelpboards.com/threads/properties-about-vectors.27360/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n\in \mathbb{N}$, $V=\mathbb{R}^n$ and $\cdot$ the standard scalar multiplication. Let $b_1, \ldots , b_k\in V$ such that $$b_i\cdot b_j=\delta_{ij}$$
Show that $b_1, \ldots , b_k$ are linear independent and that $k\leq n$.
Let $k=n$. Show that $B=(b_1, \ldots , b_n)$ is a basis of $V$ and it holds that...
Properties about vectors]]>4Show properties of matrix AMon, 27 Apr 2020 15:43:41 +0000
https://mathhelpboards.com/threads/show-properties-of-matrix-a.27308/
https://mathhelpboards.com/threads/show-properties-of-matrix-a.27308/invalid@example.com (mathmari)mathmariHey!!
We have the matrix $A=\frac{1}{3}\begin{pmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{pmatrix}$. Show that there is an unit vector $v_1$, such that $A=I-2v_1v_1^T$.
We consider an orthogonal matrix $Q=\begin{pmatrix}v_1 & v_2 & v_3\end{pmatrix}$. Show that $Q^TAQ=\begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$.
Using the last relation show that $\det A=1$.
For the first part I have done the following:
Let $v_1=\begin{pmatrix}x_1 & x_2 &...
Show properties of matrix A]]>11Test for compatibility of equations - Determinant |A b|Mon, 27 Apr 2020 11:44:51 +0000
https://mathhelpboards.com/threads/test-for-compatibility-of-equations-determinant-a-b.27306/
https://mathhelpboards.com/threads/test-for-compatibility-of-equations-determinant-a-b.27306/invalid@example.com (mathmari)mathmariHey!!
Let $Ax=b$ be a system of linear equations, where the number of equations is by one larger than the number of unknown variables, so the matrix $A$ is of full column rank.
Why can the test for combatibility of equations use the criterion of the determinant $|A \ b|$ ? ]]>4Show that there are vectors to get a basisFri, 24 Apr 2020 22:55:08 +0000
https://mathhelpboards.com/threads/show-that-there-are-vectors-to-get-a-basis.27293/
https://mathhelpboards.com/threads/show-that-there-are-vectors-to-get-a-basis.27293/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq k,m,n\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $U$ a subspace of $V$ with $\dim_{\mathbb{R}}U=m$. Let $u_1, \ldots , u_k\in U$ be linear independent. Show that there are vectors $u_{k+1}, \ldots , u_m\in U$ such that $(u_1, \ldots , u_m)$ is a basis of $U$.
Hint: Use the following two statements. Convince that $\ell:=m-k\geq 0$ and use induction on $\ell$.
Statement 1:
Let $1\leq k\in \mathbb{N}$ and $v_1, \ldots v_k, w\in V$.
$1\leq m\in...
Show that there are vectors to get a basis]]>9Kernel of linear mapFri, 24 Apr 2020 22:28:52 +0000
https://mathhelpboards.com/threads/kernel-of-linear-map.27296/
https://mathhelpboards.com/threads/kernel-of-linear-map.27296/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n,m\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $(b_1, \ldots , b_n)$ a basis of $V$. Let $W:=\mathbb{R}^m$ and let $\phi:V\rightarrow W$ be a linear map.
Show that $$\ker \phi =\left \{\sum_{i=1}^n\lambda_ib_i\mid \begin{pmatrix}\lambda_1\\ \vdots \\ \lambda_n\end{pmatrix}\in \textbf{L}(\phi (b_1), \ldots , \phi (b_n))\right \}$$
I have done the following:
Let $v\in V$. Since $(b_1, \ldots , b_n)$ is a basis of $V$, we have that...
Kernel of linear map]]>5introduction to linear algebraSun, 19 Apr 2020 19:28:36 +0000
https://mathhelpboards.com/threads/introduction-to-linear-algebra.27263/
https://mathhelpboards.com/threads/introduction-to-linear-algebra.27263/invalid@example.com (abs)absprove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.]]>5Method of least square: initial position & velocitySat, 11 Apr 2020 13:24:30 +0000
https://mathhelpboards.com/threads/method-of-least-square-initial-position-velocity.27225/
https://mathhelpboards.com/threads/method-of-least-square-initial-position-velocity.27225/invalid@example.com (mathmari)mathmariHey!!
A point is moving linearly with constant velocity $v$ and the movement is $x=a+vt$.
The below information is given:
Find the initial position $a$ and the velocity using the method of least square.
Could you give me a hint how we use this method here? Couldn't we use the data of the matrix and get a $2\times 2$ linear system? ]]>9Projection matrixSat, 11 Apr 2020 13:12:46 +0000
https://mathhelpboards.com/threads/projection-matrix.27227/
https://mathhelpboards.com/threads/projection-matrix.27227/invalid@example.com (mathmari)mathmariHey!!
We have the vectors $\displaystyle{a_k=\begin{pmatrix}\cos \frac{k\pi}{3} \\ \sin \frac{k\pi}{3}\end{pmatrix}, \ k=0, 1, \ldots , 6}$. Let $P_k$ be the projection matrix onto $a_k$.
Calculate $P_6P_5P_4P_3P_2P_1a_0$.
Are the elements of the projection matrix defined as $P_{ij}=\frac{a_ij_j}{a\cdot a}$ ? ]]>4System no/infinitely many solution(s)Sat, 11 Apr 2020 13:11:21 +0000
https://mathhelpboards.com/threads/system-no-infinitely-many-solution-s.27224/
https://mathhelpboards.com/threads/system-no-infinitely-many-solution-s.27224/invalid@example.com (mathmari)mathmariHey!!
We have the matrix $A=\begin{pmatrix}1 & -1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{pmatrix}$ and the vectors $b_1=\begin{pmatrix}1 \\ 0 \\1\end{pmatrix}$ and $b_2=\begin{pmatrix}-1 \\ 1 \\2\end{pmatrix}$.
Check if the system $Ax=b_i$ for $i\in \{1,2\}$ has a solution.
If the system is impossible find the solution that we get if the vector $b_i$ is projected onto the column space.
If the system has infinitely many solutions find the solution that belongs to the row space.
I have...
System no/infinitely many solution(s)]]>15Show that A is identicalSat, 11 Apr 2020 06:27:50 +0000
https://mathhelpboards.com/threads/show-that-a-is-identical.27223/
https://mathhelpboards.com/threads/show-that-a-is-identical.27223/invalid@example.com (evinda)evindaHello!!!
I want to prove that if a $m \times m$ matrix $A$ has rank $m$ and satisfies the condition $A^2=A$ then it will be identical.
$A^2=A \Rightarrow A^2-A=0 \Rightarrow A(A-I)=0$.
From this we get that either $A=0$ or $A=I$.
Since $A$ has rank $m$, it follows that it has $m$ non-zero rows, and so it cannot be $0$.
Thus $A=I$. Is everything right? Or could something be improved? ]]>6prove that CnXCm isomorphic to Cgcd(m,n)XClcm(m,n)Mon, 06 Apr 2020 14:08:43 +0000
https://mathhelpboards.com/threads/prove-that-cnxcm-isomorphic-to-cgcd-m-n-xclcm-m-n.27193/
https://mathhelpboards.com/threads/prove-that-cnxcm-isomorphic-to-cgcd-m-n-xclcm-m-n.27193/invalid@example.com (Alex224)Alex224Hi, I have a problem in my homework that I am stuck with.
Let there be two natural numbers n,m.
let there be d = greatest common divisor of m and n - gcd(m,n)
and l = least common multiple of m and n - lcm(m,n)
I need to prove that CnXCm isomprphic to ClXCd (Cm Cn Cl Cd are all cyclic groups)
I have tried to see what happens if I look at m and n as products of prime numbers but I am kind of stuck around that idea without knowing where to take it.
I also think I should use the fact that...
prove that CnXCm isomorphic to Cgcd(m,n)XClcm(m,n)]]>3Find the matrix AThu, 26 Mar 2020 18:09:06 +0000
https://mathhelpboards.com/threads/find-the-matrix-a.27160/
https://mathhelpboards.com/threads/find-the-matrix-a.27160/invalid@example.com (evinda)evindaHello!!!
The general solution of the system $Ax=\begin{bmatrix}
1\\
3
\end{bmatrix}$ is $x=\begin{bmatrix}
1\\
0
\end{bmatrix}+ \lambda \begin{bmatrix}
0\\
1
\end{bmatrix}$. I want to find the matrix $A$.
Let $A=\begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix}$.
$$\begin{bmatrix}
a_{11} &...
Find the matrix A]]>12Complex-Linear Matrices and C-Linear Transformations ... Tapp, Propostion 2.5 ... ...Mon, 02 Mar 2020 04:51:54 +0000
https://mathhelpboards.com/threads/complex-linear-matrices-and-c-linear-transformations-tapp-propostion-2-5.27108/
https://mathhelpboards.com/threads/complex-linear-matrices-and-c-linear-transformations-tapp-propostion-2-5.27108/invalid@example.com (Peter)PeterI am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.
I am currently focused on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".
I need help in fully understanding Tapp's Proposition 2.5.
Proposition 2.5 and some comments following it read as follows:
My questions are as follows:
Question 1
In the above text from Tapp we read the following:
" ... ... Suppose that...
Complex-Linear Matrices and C-Linear Transformations ... Tapp, Propostion 2.5 ... ...]]>0Matrices and Linear Transformations ... Armstrong, Ch. 9 and Tapp Ch. 1, Section 5 ...Sun, 01 Mar 2020 04:19:53 +0000
https://mathhelpboards.com/threads/matrices-and-linear-transformations-armstrong-ch-9-and-tapp-ch-1-section-5.27098/
https://mathhelpboards.com/threads/matrices-and-linear-transformations-armstrong-ch-9-and-tapp-ch-1-section-5.27098/invalid@example.com (Peter)PeterAt the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following:
" ... ... Each matrix \(\displaystyle A\) in this group determines an invertible linear transformation \(\displaystyle f_A: \mathbb{R} \to \mathbb{R}\) defined by \(\displaystyle f_A(x) = x A^t\) ... ... "
I know that one may define entities how one wishes ... but why does Armstrong define \(\displaystyle f\) in terms of the transpose of \(\displaystyle A\) rather than just simply...
Matrices and Linear Transformations ... Armstrong, Ch. 9 and Tapp Ch. 1, Section 5 ...]]>2Complex-Linear Matrices and C-Linear Transformations ... Tapp, Propostion 2.4 ... ...Sun, 01 Mar 2020 04:14:24 +0000
https://mathhelpboards.com/threads/complex-linear-matrices-and-c-linear-transformations-tapp-propostion-2-4.27106/
https://mathhelpboards.com/threads/complex-linear-matrices-and-c-linear-transformations-tapp-propostion-2-4.27106/invalid@example.com (Peter)PeterI am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.
I am currently focused on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".
I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4.
Proposition 2.4 and some comments following it read as follows:
In the remarks following Proposition 2.4 we read the following:
Vector calculations - Geogebra]]>3Determine the cycle decomposition of the permutationsFri, 21 Feb 2020 05:43:52 +0000
https://mathhelpboards.com/threads/determine-the-cycle-decomposition-of-the-permutations.27062/
https://mathhelpboards.com/threads/determine-the-cycle-decomposition-of-the-permutations.27062/invalid@example.com (mathmari)mathmariHey!!
We have the following permutations in $\text{Sym}(14)$ :
1. Determine the cycle decomposition of $\pi_1, \pi_2, \pi_3$.
2. Determine $\pi_1^{-1}, \pi_2^{-1}, \pi_3^{-1}$.
3. Determine $\pi_4=\pi_1\circ \pi_2$, $\pi_5=\pi_2\circ\pi_3$, $\pi_6=\pi_2\circ\pi_1$.
4. Determine the signum of $\pi_1, \pi_2...
Determine the cycle decomposition of the permutations]]>28Condition for a unique solution of matrix A that have infinite solutions and optimisation of tr(A)Sat, 15 Feb 2020 18:48:51 +0000
https://mathhelpboards.com/threads/condition-for-a-unique-solution-of-matrix-a-that-have-infinite-solutions-and-optimisation-of-tr-a.27083/
https://mathhelpboards.com/threads/condition-for-a-unique-solution-of-matrix-a-that-have-infinite-solutions-and-optimisation-of-tr-a.27083/invalid@example.com (zeeshas901)zeeshas901Hello!
I am new here, and I need (urgent) help regarding the following question:
Let $\boldsymbol{A}_{(n\times n)}=[a_{ij}]$ be a square matrix such that the sum of each row is 1 and $a_{ij}\ge0$$(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$ are unknown. Suppose that $\boldsymbol{b}_{1}=[b_{11}~b_{12}\dots b_{1n}]$ and $\boldsymbol{b}_{2}=[b_{21}~b_{22}\dots b_{2n}]$ are known row vectors of proportions such that $$\boldsymbol{b}_{1}\boldsymbol{A}_{(n\times n)}=\boldsymbol{b}_{2},$$ where...
1. Let $1\leq n\in \mathbb{N}$ and $\pi\in \text{Sym}(n)$. For $1\leq k\in \mathbb{N}$ we define $\pi^{-k}:=\left (\pi^n\right )^{-1}$.
Show for all $k,\ell\in \mathbb{Z}$ the equation $\pi^k\circ \pi^{\ell}=\pi^{k+\ell}$.
2. Let $1\leq n\in \mathbb{N}$. Show that $\pi^{n!}=\text{id}$ for all $\pi\in \text{Sym}(n)$.
Do we show both statements using induction? ]]>5Which is the geometric interpretation?Fri, 14 Feb 2020 02:39:48 +0000
https://mathhelpboards.com/threads/which-is-the-geometric-interpretation.27067/
https://mathhelpboards.com/threads/which-is-the-geometric-interpretation.27067/invalid@example.com (mathmari)mathmariHey!!
Which is the geometric interpretation of the following maps?
$$v\mapsto \begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&-1\end{pmatrix}v$$ and $$v\mapsto \begin{pmatrix} 1& 0&0\\0&\frac{1}{2} &-\frac{\sqrt{3}}{2}\\ 0&\frac{\sqrt{3}}{2}&\frac{1}{2}\end{pmatrix}v$$]]>4Show the properties of σThu, 13 Feb 2020 08:33:15 +0000
https://mathhelpboards.com/threads/show-the-properties-of-%CF%83.27065/
https://mathhelpboards.com/threads/show-the-properties-of-%CF%83.27065/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n\in \mathbb{N}$. For $0_{\mathbb{R}^n}\neq x\in \mathbb{R}^n$ we define the map $$\sigma_x:\mathbb{R}^n\rightarrow \mathbb{R}^n, \ v\mapsto v-2\frac{x\cdot v}{x\cdot x}x$$
Show that:
The map is linear.
It holds that $\sigma_x\in \text{Sym}(\mathbb{R}^n)$ and $\sigma_x=\sigma_x^{-1}$.
For all $v,w\in \mathbb{R}$ it holds that $v\cdot w=\sigma_x(v)\cdot \sigma_x(w)$.
Let $n=2$. Determine a vector $v\in \mathbb{R}^2$ such that...
Show the properties of σ]]>11Interpretation of mapsMon, 10 Feb 2020 08:51:39 +0000
https://mathhelpboards.com/threads/interpretation-of-maps.27061/
https://mathhelpboards.com/threads/interpretation-of-maps.27061/invalid@example.com (mathmari)mathmariHey!!
Let $\phi_1:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $v\mapsto \begin{pmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}\cdot v$, $\phi_2:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $v\mapsto \frac{1}{3}\begin{pmatrix}2 & - 1& - 1\\ - 1& 2 & -1\\ - 1&-1 & 2\end{pmatrix}\cdot v$ which describes the shadow of the sun on a plane.
Give the direction of the sun by giving a vector. How do we find this vector?
To give the equation of the plane do we have to find the image of...
Interpretation of maps]]>19Calculate the determinantsMon, 10 Feb 2020 07:38:45 +0000
https://mathhelpboards.com/threads/calculate-the-determinants.27060/
https://mathhelpboards.com/threads/calculate-the-determinants.27060/invalid@example.com (mathmari)mathmariHey!!
For that do we use the Laplace expansion theorem or can we transform these matrices firstly in echelon form and...
Calculate the determinants]]>13Determine the solution set of the system using the echelon formSat, 08 Feb 2020 15:09:45 +0000
https://mathhelpboards.com/threads/determine-the-solution-set-of-the-system-using-the-echelon-form.27053/
https://mathhelpboards.com/threads/determine-the-solution-set-of-the-system-using-the-echelon-form.27053/invalid@example.com (mathmari)mathmariHey!!
I applied the Gauss algorithm to get the echelon form of the matrix $a$ :
\begin{align*}\begin{pmatrix}2 & 1 & 0 & 5 \\ 1 & 0 & 1 & 1 \\ 4 & 1 &2 & 7\end{pmatrix} & \ \overset{R_2:R_2-\frac{1}{2}\cdot R_1}{\longrightarrow}...
Determine the solution set of the system using the echelon form]]>2How is M defined?Tue, 04 Feb 2020 19:11:20 +0000
https://mathhelpboards.com/threads/how-is-m-defined.27028/
https://mathhelpboards.com/threads/how-is-m-defined.27028/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n,m\in \mathbb{N}$ and let $\phi, \psi:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be linear maps. Let $\lambda\in \mathbb{R}$.
Show the following:
$M(\phi +\psi )=M(\phi )+M(\psi )$
$M(\lambda \phi )=\lambda M(\phi )$
What exactly is $M$, it is not defined in this exercise? Is it a matrix? ]]>10Symmetry map and rotationTue, 04 Feb 2020 18:07:54 +0000
https://mathhelpboards.com/threads/symmetry-map-and-rotation.27029/
https://mathhelpboards.com/threads/symmetry-map-and-rotation.27029/invalid@example.com (mathmari)mathmariHey!!
Let $F\subseteq \mathbb{R}^2$. A map $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ is called symmetry map of $F$, if $\pi(F)=F$. A symmetry map of $F$ is a map where the figures $F$ and $\pi (F)$ are congruent.
Let $\pi_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $x\mapsto \begin{pmatrix}0 &-1 \\ 1 & 0\end{pmatrix}\cdot x$ and $\pi_2:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $x\mapsto \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\cdot x$ be symmetry maps of the below square...
Symmetry map and rotation]]>17Rotations, Reflections and TranslationsTue, 04 Feb 2020 18:04:49 +0000
https://mathhelpboards.com/threads/rotations-reflections-and-translations.27039/
https://mathhelpboards.com/threads/rotations-reflections-and-translations.27039/invalid@example.com (mathmari)mathmariHey!!
Let $ \tau_v: \mathbb{R}^2 \rightarrow \mathbb{R}^2, \ \ x \mapsto x + v $ be the shift by the vector $ v \in \mathbb{R}^2 $.
Let $ \sigma_a: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be the reflection on the straight line through the origin, where $ a $ describes the angle between the straight line and the positive $ x $ axis.
One can show that the mapping $\sigma_a$ is linear.
Determine $\sigma_a\begin{pmatrix} 1\\ 0\end{pmatrix}$ and $\sigma_a\begin{pmatrix} 0\\...
Rotations, Reflections and Translations]]>4Define matrix to get a row operation of type 1Tue, 04 Feb 2020 18:03:52 +0000
https://mathhelpboards.com/threads/define-matrix-to-get-a-row-operation-of-type-1.27043/
https://mathhelpboards.com/threads/define-matrix-to-get-a-row-operation-of-type-1.27043/invalid@example.com (mathmari)mathmariHey!!
Define matrix to get a row operation of type 1]]>2The permutation induces on the setSat, 01 Feb 2020 18:25:06 +0000
https://mathhelpboards.com/threads/the-permutation-induces-on-the-set.27037/
https://mathhelpboards.com/threads/the-permutation-induces-on-the-set.27037/invalid@example.com (mathmari)mathmariHey!!
I am looking at the following exercise:
Make a sketch of a regular tetrahedron and label the corners with the numbers $1, 2, 3, 4$. For $1\leq i\leq 5$ the permutations $\pi_i \in \text{Sym} (4)$ are defined as follows:
\begin{align*}&\pi_1:=\text{id} \\ &\pi_2:=(1 \ \ 2) \\ &\pi_3:=(1 \ \ 2 \ \ 3) \\ &\pi_4:=(1 \ \ 2)\ (3 \ \ 4) \\ &\pi_5:=(1 \ \ 2 \ \ 3 \ \ 4)\end{align*}
For $1 \leq i \leq 5$, check whether there is a symmetry of the tetrahedron that the...
The permutation induces on the set]]>23USMA linear systems of equation solve for x y zTue, 21 Jan 2020 21:18:06 +0000
https://mathhelpboards.com/threads/usma-linear-systems-of-equation-solve-for-x-y-z.26950/
https://mathhelpboards.com/threads/usma-linear-systems-of-equation-solve-for-x-y-z.26950/invalid@example.com (karush)karush
ok lots of options to solve this but I would start by $R3-R1\to R3$
if I remember correctly if get a diagonal of ones and the rest zeros in A we will have B from Ax=B
$\tiny{USMA = United \,States\, Military\, Academy}$
]]>4Show that it is a subspaceSat, 11 Jan 2020 16:18:01 +0000
https://mathhelpboards.com/threads/show-that-it-is-a-subspace.26988/
https://mathhelpboards.com/threads/show-that-it-is-a-subspace.26988/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq m, n\in \mathbb{N}$, let $\phi :\mathbb{R}^n\rightarrow \mathbb{R}^m$ a linear map and let $U\leq_{\mathbb{R}}\mathbb{R}^n$, $W\leq_{\mathbb{R}}\mathbb{R}^m$ be subspaces.
I want to show that:
$\phi (U)$ is subspace of $\mathbb{R}^m$.
$\phi^{-1} (W)$ is subspace of $\mathbb{R}^n$.
I have done the following:
We have that $\phi (U)=\{\phi (u) \mid u\in U\}$.
Since $U$ is a subspace we have that $0\in U$. Therefore $\phi...
Show that it is a subspace]]>10Decide if the sets are subspaces or affine subspacesSat, 07 Dec 2019 15:04:05 +0000
https://mathhelpboards.com/threads/decide-if-the-sets-are-subspaces-or-affine-subspaces.26774/
https://mathhelpboards.com/threads/decide-if-the-sets-are-subspaces-or-affine-subspaces.26774/invalid@example.com (mathmari)mathmariHey!!
Decide if the sets are subspaces or affine subspaces]]>3Basis of spanSat, 07 Dec 2019 15:01:33 +0000
https://mathhelpboards.com/threads/basis-of-span.26806/
https://mathhelpboards.com/threads/basis-of-span.26806/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n\in \mathbb{N}$ and $v_1, \ldots , v_k\in \mathbb{R}^n$. Show that there exist $w_1, \ldots , w_m\in \{v_1, \ldots , v_k\}$ such that $(w_1, \ldots , w_m)$ is a basis of $\text{Lin}(v_1, \ldots , v_k)$.
I have done the following:
A basis of $\text{Lin}(v_1, \ldots , v_k)$ is a linearly independent set of vectors of $\{v_1, \ldots , v_k\}$.
So let $\{w_1, \ldots , w_m\}\subseteq \{v_1, \ldots , v_k\}$ be a linearly independent set.
$\text{Lin}(v_1, \ldots ...
Basis of span]]>3Statements with determinantSat, 07 Dec 2019 15:00:07 +0000
https://mathhelpboards.com/threads/statements-with-determinant.26822/
https://mathhelpboards.com/threads/statements-with-determinant.26822/invalid@example.com (mathmari)mathmariHey!!
We have the matrix $A=\begin{pmatrix}a_1 & b_1 \\ a_2 & b_2\end{pmatrix}$.
We consider the vectors $\vec{v}:=A\vec{e}_1$ and $\vec{w}:=A\vec{e}_2$.
Justify geometrically, why the area of the parallelogram spanned by $\vec{v}$ and $\vec{w}$ is equal to $\det A$.
Calculate the determinant of the matrix that we get from $A$ if we swap the two columns. (This is related to whether the orientation of the parallelogram is equal to the orientation of $\mathbb{R}^2$...
Statements with determinant]]>15Relationship between three planesTue, 26 Nov 2019 05:11:00 +0000
https://mathhelpboards.com/threads/relationship-between-three-planes.26811/
https://mathhelpboards.com/threads/relationship-between-three-planes.26811/invalid@example.com (mathmari)mathmariHey!!
Let an arbitrary linear system of $3$ equations and $3$ variables be given. There are $4$ cases how the planes can be related.
Describe these $4$ cases graphically and describe the set of solutions in each case.
I have done the following:
If the three equations are linearly independent, then the system has a single solution.
In this case the three planes described by the equations intersect, since they are neither parallel nor identical.
The intersection of these three...
Relationship between three planes]]>5Norms for a Linear Transformation ... Browder, Lemma 8.4 ...Sun, 24 Nov 2019 11:35:39 +0000
https://mathhelpboards.com/threads/norms-for-a-linear-transformation-browder-lemma-8-4.26812/
https://mathhelpboards.com/threads/norms-for-a-linear-transformation-browder-lemma-8-4.26812/invalid@example.com (Peter)PeterI am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...
I need some help in fully understanding Lemma 8.4 ...
Lemma 8.4 reads as follows:
In the above proof of Lemma 8.4 by Browder we read the following:
" ... ... On the other hand since \(\displaystyle \sum_{ j = 1 }^m (...\)
Norms for a Linear Transformation ... Browder, Lemma 8.4 ...]]>1Set of all Linear Transformations, L(R^n, R^m) ... Remarks by Browder, Section 8.1, Page 179 ... ...Wed, 20 Nov 2019 04:02:38 +0000
https://mathhelpboards.com/threads/set-of-all-linear-transformations-l-r-n-r-m-remarks-by-browder-section-8-1-page-179.26788/
https://mathhelpboards.com/threads/set-of-all-linear-transformations-l-r-n-r-m-remarks-by-browder-section-8-1-page-179.26788/invalid@example.com (Peter)PeterI am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am reading Chapter 8: Differentiable Maps ... ... and am currently focused on Section 8.1 Linear Algebra ... ...
I need some help in order to fully understand some remarks by Browder in Section 8.1, page 179 regarding the set of all linear transformations, \(\displaystyle \mathscr{L} ( \mathbb{R^n, R^m} )\) ... ...
The relevant statements by Browder follow Definition 6.10 and read as follows...
Let $1\leq n,k\in \mathbb{N}$ and let $v_1, \ldots , v_k\in \mathbb{R}^k$. Show that:
Let $w\in \text{Lin}(v_1, \ldots , v_k)$. Then it holds that $\text{Lin}(v_1, \ldots , v_k)=\text{Lin}(v_1, \ldots , v_k,w)$.
Let $v_1, \ldots , v_k$ be linearly dependent. Thn there is a $1\leq i\leq k$ and $\lambda_1, \ldots , \lambda_k$ such that $v_i=\lambda_1v_1+\ldots +\lambda_{i-1}v_{i-1}+\lambda_{i+1}v_{i+1}+\ldots +\lambda_nk_n$.
Let $i_1, \ldots i_k\in...
Statements about span]]>3Are the vectors linearly independent?Sat, 16 Nov 2019 17:20:29 +0000
https://mathhelpboards.com/threads/are-the-vectors-linearly-independent.26775/
https://mathhelpboards.com/threads/are-the-vectors-linearly-independent.26775/invalid@example.com (mathmari)mathmariHey!!
We have that the vectrs $\vec{v},\vec{w}, \vec{u}$ are linearly independent.
I want to check if the pairs
$\vec{v}, \vec{v}+\vec{w}$
$\vec{v}+\vec{u}$, $\vec{w}+\vec{u}$
$\vec{v}+\vec{w}$, $\vec{v}-\vec{w}$
are linearly indeendent or not.
Since $\vec{v}, \vec{w}, \vec{u}$ are linearly independet it holds that $\lambda_1\vec{v}+\lambda_2\vec{w}+\lambda_3\vec{u}=0 \Rightarrow \lambda_1=\lambda_2=\lambda_3=0$ ($\star$).
We have the following...
Are the vectors linearly independent?]]>8Subset that satisfies all but one axioms of subspacesWed, 13 Nov 2019 20:44:17 +0000
https://mathhelpboards.com/threads/subset-that-satisfies-all-but-one-axioms-of-subspaces.26752/
https://mathhelpboards.com/threads/subset-that-satisfies-all-but-one-axioms-of-subspaces.26752/invalid@example.com (mathmari)mathmariHey!!
I want to find subsets $S$ of $\mathbb{R}^2$ such that $S$ satisfies all but one axioms of subspaces.
A subset that doesn't satisfy the first axiom: We have to find a subset that doesn't contain the zero vector. Is this for example $\left \{\begin{pmatrix}x \\ y\end{pmatrix} : x,y>0\right \}$ ?
A subset that doesn't satisfy the second axiom: We have to find a subset that doesn't contain the sum of two vectors of $S$. Could you give me an example for that?
A subset that...
Subset that satisfies all but one axioms of subspaces]]>16Intersection of two spansWed, 13 Nov 2019 19:36:12 +0000
https://mathhelpboards.com/threads/intersection-of-two-spans.26716/
https://mathhelpboards.com/threads/intersection-of-two-spans.26716/invalid@example.com (mathmari)mathmariHey!!
I want to calculate the intersection of the spans $\text{Lin}(v_1, v_2, v_3)\cap \text{Lin}(w_1, w_2)$.
We have \begin{align*}&\text{Lin}(v_1, v_2, v_3)=\left...
Intersection of two spans]]>8Are these sets subspaces?Sat, 02 Nov 2019 11:09:01 +0000
https://mathhelpboards.com/threads/are-these-sets-subspaces.26718/
https://mathhelpboards.com/threads/are-these-sets-subspaces.26718/invalid@example.com (mathmari)mathmariHey!!
We have the following subsets:
\begin{align*}&U_1:=\left \{\begin{pmatrix}x \\ y\end{pmatrix} \mid x^2+y^2\leq 4\right \} \subseteq \mathbb{R}^2\\ &U_2:=\left \{\begin{pmatrix}2a \\ -a\end{pmatrix} \mid a\in \mathbb{R}\right \} \subseteq \mathbb{R}^2 \\ &U_3:=\left \{\begin{pmatrix}x \\ y \\ z\end{pmatrix} \mid y=0\right \}\subseteq \mathbb{R}^3 \\ &U_4:=\left \{\begin{pmatrix}x \\ y \\ z\end{pmatrix} \mid y=1\right \}\subseteq \mathbb{R}^3\end{align*}
I want to sketch these...
Are these sets subspaces?]]>1Rewriting vectors in different coordinatesThu, 31 Oct 2019 16:33:33 +0000
https://mathhelpboards.com/threads/rewriting-vectors-in-different-coordinates.26705/
https://mathhelpboards.com/threads/rewriting-vectors-in-different-coordinates.26705/invalid@example.com (goohu)goohuLets say you have a vector in spherical coordinates; how do you rewrite this vector into a cartesian one and vice versa?
Im fine with rewriting coordinates but vectors have got me confused. I've tried digging through info online but I couldn't find any good examples.
In the following task, Ive found the gradient of the Point P. I'm stuck at the last step trying to find the cartesian coordinates.
We should use the following equations to transform...
Rewriting vectors in different coordinates]]>0concatenation of listsTue, 29 Oct 2019 11:36:43 +0000
https://mathhelpboards.com/threads/concatenation-of-lists.26701/
https://mathhelpboards.com/threads/concatenation-of-lists.26701/invalid@example.com (Kenan)KenanHallo guys,
I'm looking for your help . Here is a question from an Assigment, that i should tomorrow gave.
Explain that the concatenation of lists is associative but not commutative and not idempotent. (In this respect, there is one thing in common, however, two differences from the union.) Use as a symbol for the concatenation
K..L for lists K and L.
Thanks in Advance!]]>1Prove the statements : Vectors/MatricesMon, 28 Oct 2019 20:39:26 +0000
https://mathhelpboards.com/threads/prove-the-statements-vectors-matrices.26694/
https://mathhelpboards.com/threads/prove-the-statements-vectors-matrices.26694/invalid@example.com (mathmari)mathmariHey!!
Let $1\leq n\in \mathbb{N}$.
Prove that for all $v\in \mathbb{R}^n$ it holds that $v+0_{\mathbb{R}^n}=v=0_{\mathbb{R}^n}+v$.
Prove that for all $\lambda\in \mathbb{R}$ and $v,w\in \mathbb{R}$ it holds that $\lambda (v+w)=\lambda v+\lambda w$.
Let $M_2(\mathbb{R}):=\left \{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\mid a, b, c, d\in \mathbb{R}\right \}$ the set of all $2\times 2$-matrices over $\mathbb{R}$. We define also the multiplication on that set as...
Prove the statements : Vectors/Matrices]]>5Prove properties of distanceSat, 26 Oct 2019 15:55:03 +0000
https://mathhelpboards.com/threads/prove-properties-of-distance.26688/
https://mathhelpboards.com/threads/prove-properties-of-distance.26688/invalid@example.com (mathmari)mathmariHey!!
Let $v, w\in \mathbb{R} ^n$ and let $V, W\subseteq \mathbb{R} ^n$.
I want to show the following properties :
$d(u.,w)=0\iff u=v$
$d(V, W) =0\iff V\cap W\neq \emptyset$
I have done the following:
$d(u, w) =0\iff |u-w|=0\iff u-w=0\iff u=w$
Or do we have to do more steps?
$$$$
$d(V, W) =0\iff \min \{d(v, w) \} =0$ this means that there exists $v$ and $w$ such that $d(v, w) =0$ and from the previous one it follows that $v=w$ which means...
Prove properties of distance]]>10Characteristics of ringsWed, 16 Oct 2019 19:51:53 +0000
https://mathhelpboards.com/threads/characteristics-of-rings.26664/
https://mathhelpboards.com/threads/characteristics-of-rings.26664/invalid@example.com (fredpeterson57)fredpeterson57Context: Let R be a unital ring. The characteristic of R is the smallest positive integer n such that $n\cdot 1=0$. If no such n exists, we say R has characteristic 0. We denote the characteristic of a ring by char(R).
I'm particularly lost as to how to prove the following propositions:
(a) Every unital ring of characteristic zero is infinite (I'm thinking of using a proof by contradiction for this, but I have no idea how)
(b) The characteristic of an integral domain is either 0 or prime...
Characteristics of rings]]>14 problems regarding automorphisms/homomorphismsTue, 08 Oct 2019 19:50:07 +0000
https://mathhelpboards.com/threads/4-problems-regarding-automorphisms-homomorphisms.26628/
https://mathhelpboards.com/threads/4-problems-regarding-automorphisms-homomorphisms.26628/invalid@example.com (AutGuy98)AutGuy98Hey guys,
I have some more problems that I need help with figuring out what to do. The second (and final) one is divided into 4 mini-problems, sub-sections, whatever you would like to call them. It asks:
(a) Show that the set of automorphisms of a group G, denoted by Aut(G), is a group under the usual composition of functions.
(b) Let G be a group and $g\in G$. Define a map $\psi_g:G\to G$ as follows: for any $h\in G, \psi_g(h)=ghg^{-1}$. Show that $\psi_g$ is an automorphism.
(c) Show...
4 problems regarding automorphisms/homomorphisms]]>54 part problem regarding homomorphismsMon, 07 Oct 2019 19:27:46 +0000
https://mathhelpboards.com/threads/4-part-problem-regarding-homomorphisms.26627/
https://mathhelpboards.com/threads/4-part-problem-regarding-homomorphisms.26627/invalid@example.com (AutGuy98)AutGuy98Hey guys,
I have some more problems that I need help with figuring out what to do. The first one is divided into 4 mini-problems, sub-sections, whatever you would like to call them. It asks:
(a) Show that (Z/4Z,+) is not isomorphic to ((Z/2Z) x (Z/2Z),+). Find a homomorphism from (Z/4Z,+) to ((Z/2Z) x (Z/2Z),+).
(b) Let G and H be groups and let $\phi:G\to H$ be a homomorphism. Show that $ker(\phi)$ is a normal subgroup of G and that $\phi(G)$ is a subgroup of H.
(c) Find a pair of groups...
4 part problem regarding homomorphisms]]>0List all the subgroups H of C_(12)Wed, 18 Sep 2019 00:21:54 +0000
https://mathhelpboards.com/threads/list-all-the-subgroups-h-of-c_-12.26554/
https://mathhelpboards.com/threads/list-all-the-subgroups-h-of-c_-12.26554/invalid@example.com (AutGuy98)AutGuy98Hey guys,
Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in light of that, I actually have some more questions for an exercise set that I have to do for one of my classes and I'm really unsure how to begin doing them. There are four of them and they all...
List all the subgroups H of C_(12)]]>5Find n such that the group of the n-th roots of unity has exactly 6 generatorsTue, 17 Sep 2019 21:04:21 +0000
https://mathhelpboards.com/threads/find-n-such-that-the-group-of-the-n-th-roots-of-unity-has-exactly-6-generators.26555/
https://mathhelpboards.com/threads/find-n-such-that-the-group-of-the-n-th-roots-of-unity-has-exactly-6-generators.26555/invalid@example.com (AutGuy98)AutGuy98Hey guys,
Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in light of that, I actually have some more questions for an exercise set that I have to do for one of my classes and I'm really unsure how to begin doing them. There are four of them and they all...
Find n such that the group of the n-th roots of unity has exactly 6 generators]]>3Prove that the 12-th roots of unity in C form a cyclic groupTue, 17 Sep 2019 20:18:21 +0000
https://mathhelpboards.com/threads/prove-that-the-12-th-roots-of-unity-in-c-form-a-cyclic-group.26553/
https://mathhelpboards.com/threads/prove-that-the-12-th-roots-of-unity-in-c-form-a-cyclic-group.26553/invalid@example.com (AutGuy98)AutGuy98Hey guys,
Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in light of that, I actually have some more questions for an exercise set that I have to do for one of my classes and I'm really unsure how to begin doing them. There are four of them and they all...
Prove that the 12-th roots of unity in C form a cyclic group]]>3Prove Lagrange’s Theorem for left cosetsTue, 17 Sep 2019 20:01:09 +0000
https://mathhelpboards.com/threads/prove-lagrange%E2%80%99s-theorem-for-left-cosets.26552/
https://mathhelpboards.com/threads/prove-lagrange%E2%80%99s-theorem-for-left-cosets.26552/invalid@example.com (AutGuy98)AutGuy98Hey guys,
Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in light of that, I actually have some more questions for an exercise set that I have to do for one of my classes and I'm really unsure how to begin doing them. There are four of them and they all...
Prove Lagrange’s Theorem for left cosets]]>3Proving parallel lines using points and vectorsThu, 12 Sep 2019 16:23:30 +0000
https://mathhelpboards.com/threads/proving-parallel-lines-using-points-and-vectors.26531/
https://mathhelpboards.com/threads/proving-parallel-lines-using-points-and-vectors.26531/invalid@example.com (algebruh)algebruhHey, this is a problem given to me by my prof for an assignment, and the TAs at my tutorials haven't been much help. Was wondering where to go with this question.
Also, I'm a uni freshman who isn't used to the whole concept of proofs, and a lot of what my profs say seem to be a slew of symbols and numbers before they even define anything, but I do the textbook readings and can comprehend those fairly easily. Was anyone on here's transition from high...
Proving parallel lines using points and vectors]]>1Invertiable Matrix - explaining it to children - ideas how to teachSat, 07 Sep 2019 11:28:28 +0000
https://mathhelpboards.com/threads/invertiable-matrix-explaining-it-to-children-ideas-how-to-teach.26518/
https://mathhelpboards.com/threads/invertiable-matrix-explaining-it-to-children-ideas-how-to-teach.26518/invalid@example.com (roni)roniHow can I make a curriculum to student, to explain the term Invertiable Matrix?
What would I use to explain?
[tools, whiteboard, papers. printed website pages...]
How can I order the curriculum?
What will be in the start of year and what will be next?
This new material that I need to teach in the coming year.
Thanks for responds...
-Roni]]>1Zero divisor for polynomial ringsSat, 07 Sep 2019 10:26:06 +0000
https://mathhelpboards.com/threads/zero-divisor-for-polynomial-rings.26508/
https://mathhelpboards.com/threads/zero-divisor-for-polynomial-rings.26508/invalid@example.com (Cbarker1)Cbarker1Dear Everybody,
I am having trouble with how to begin with this problem from Abstract Algebra by Dummit and Foote (2nd ed):
Let $R$ be a commutative ring with 1.
Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in $R[x]$ if and only if there is a nonzero $b\in R$ such that $bp(x)=0$.
Hint: Let $g(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots+b_0$ be a nonzero polynomial of minimal degree of such that...
Zero divisor for polynomial rings]]>1Group Ring Integral dihedral group with order 6Wed, 21 Aug 2019 07:59:17 +0000
https://mathhelpboards.com/threads/group-ring-integral-dihedral-group-with-order-6.26476/
https://mathhelpboards.com/threads/group-ring-integral-dihedral-group-with-order-6.26476/invalid@example.com (Cbarker1)Cbarker1Dear Every one,
I am having some difficulties with computing an element in the Integral dihedral group with order 6.
Some background information for what is a group ring:
A group ring defined as the following from Dummit and Foote:
Fix a commutative ring $R$ with identity $1\ne0$ and let $G=\{g_{1},g_{2},g_{3},...,g_{n}\}$ be any finite group with group operation written multiplicatively. A group ring, $RG$, of $G$ with coefficients in $R$ to be the set of all formal sum...
Group Ring Integral dihedral group with order 6]]>1Group RingsSat, 17 Aug 2019 20:59:35 +0000
https://mathhelpboards.com/threads/group-rings.26464/
https://mathhelpboards.com/threads/group-rings.26464/invalid@example.com (Cbarker1)Cbarker1Dear Everyone,
Does every theorem that holds for finite group holds for ring groups? Why or Why not?
Thanks
Cbarker1]]>35.3 Show that a square matrix with a zero row is not invertible.Fri, 09 Aug 2019 20:10:47 +0000
https://mathhelpboards.com/threads/5-3-show-that-a-square-matrix-with-a-zero-row-is-not-invertible.26441/
https://mathhelpboards.com/threads/5-3-show-that-a-square-matrix-with-a-zero-row-is-not-invertible.26441/invalid@example.com (karush)karushShow that a square matrix with a zero row is not invertible.
first a matrix has to be a square to be invertable
if
$$\det \begin{pmatrix}1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}=1$$
then $$\begin{pmatrix}
1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}^{-1}
=\begin{pmatrix}1&0&0\\ 0&1&0\\ -3&0&1
\end{pmatrix}$$
but if $r_1$ is all zeros
$$\det \begin{pmatrix}
0&0&0\\ 0&1&0\\ 3&0&0
\end{pmatrix}=0$$
then
$$\begin{pmatrix}
0&0&0\\ 0&1&0\\ 3&0&0
\end{pmatrix}^{-1}...
5.3 Show that a square matrix with a zero row is not invertible.]]>3211 For what value(s) of h is b in the plane spannedWed, 31 Jul 2019 23:58:23 +0000
https://mathhelpboards.com/threads/211-for-what-value-s-of-h-is-b-in-the-plane-spanned.26418/
https://mathhelpboards.com/threads/211-for-what-value-s-of-h-is-b-in-the-plane-spanned.26418/invalid@example.com (karush)karushFor what value(s) of $h$ is b in the plane spanned by $a_1$ and $a_2$
$$a_1=\left[\begin{array}{r} 1\\3\\ -1 \end{array}\right],
a_2=\left[\begin{array}{r} -5\\-8\\2 \end{array}\right],
b =\left[\begin{array}{r} 3\\-5\\ \color{red}{h} \end{array}\right]$$
ok this should be obvious but I don't see it..]]>5Partially unpicking coordinate rotationTue, 23 Jul 2019 15:50:11 +0000
https://mathhelpboards.com/threads/partially-unpicking-coordinate-rotation.26392/
https://mathhelpboards.com/threads/partially-unpicking-coordinate-rotation.26392/invalid@example.com (Snow)SnowI measured a vector many times, and then processed the data using a computer program. The program did a great many useful things, including rotate the coordinate system about all three axes.
I have measured values for x, y, and z along the original axes. The program helpfully gave me the values for the same vectors (u, v, and w) along the new axes. However, it did not provide the rotation angles.
I need to know how much of w acts along the original z axis. I do not need how much u and v...
ok this is already in rref and we have 3 pivots in $C_1,C_2,C_3$
so is $$RS(A)= \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix}
, \begin{bmatrix} 0\\1\\0\\0 \end{bmatrix}
, \begin{bmatrix} 0\\0\\1\\0 \end{bmatrix}$$
(b) derive dim(RS(A))...
15.3 RS(A) dim(RS(A)) dim(NS(A)) + Rank(A) = 5.]]>0State where in the ty-plane the hypotheses of Theorem 2.4.2 are satisfied.Mon, 08 Jul 2019 12:37:40 +0000
https://mathhelpboards.com/threads/state-where-in-the-ty-plane-the-hypotheses-of-theorem-2-4-2-are-satisfied.26266/
https://mathhelpboards.com/threads/state-where-in-the-ty-plane-the-hypotheses-of-theorem-2-4-2-are-satisfied.26266/invalid@example.com (karush)karush
State where in the ty-plane the hypotheses of Theorem 2.4.2 are satisfied
$\displaystyle y^\prime= \frac{t-y}{2t+5y}$
ok I don't see how this book answer was derived since not sure how to separate varibles
$2t+5y>0 \textit{ or }2t+5y<0$]]>1Problem (c) for Discrete Value RingWed, 03 Jul 2019 18:00:41 +0000
https://mathhelpboards.com/threads/problem-c-for-discrete-value-ring.26289/
https://mathhelpboards.com/threads/problem-c-for-discrete-value-ring.26289/invalid@example.com (Cbarker1)Cbarker1Problem (c) for Discrete Value Ring for a unit
I am stuck in the middle of a proof. Here is the background information from Dummit and Foote Abstract Algebra 2nd ed.:
Let $K$ be a field. A discrete valuation on $K$ on a function $\nu$: $K^{\times} \to \Bbb{Z}$ satisfying
$\nu(a\cdot b)=\nu(a)+\nu(b)$ [i.e. $\nu$ is a homomorphism from the multiplication group of nonzero elements of $K$ to $\Bbb{Z}$]
$\nu$ is surjective, and
$\nu(x+y)\ge \min{[\nu(x),\nu(y)]}$, for...
Problem (c) for Discrete Value Ring]]>11.3.11 Determine if b is a linear combinationTue, 02 Jul 2019 12:37:07 +0000
https://mathhelpboards.com/threads/1-3-11-determine-if-b-is-a-linear-combination.26304/
https://mathhelpboards.com/threads/1-3-11-determine-if-b-is-a-linear-combination.26304/invalid@example.com (karush)karushDetermine if $b$ is a linear combination of $a_1,a_2$ and $a_3$
$$a_1\left[
\begin{array}{r}
1\\-2\\0 \end{array}\right],
a_2\left[
\begin{array}{r}
0\\1\\2
\end{array}\right],
a_3\left[
\begin{array}{r}
5\\-6\\8
\end{array}\right],
b=\left[
\begin{array}{r}
2\\-1\\6
\end{array}\right]$$
(rref) augmented matrix is
$$\left[
\begin{array}{ccc|c}
1 & 0 & 5 & 2 \\
0 & 1 & 4 & 3 \\
0 & 0 & 0 & 0...
1.3.11 Determine if b is a linear combination]]>415.3 verify that dim(NS(A)) + Rank(A) = 5Sun, 23 Jun 2019 00:08:05 +0000
https://mathhelpboards.com/threads/15-3-verify-that-dim-ns-a-rank-a-5.26286/
https://mathhelpboards.com/threads/15-3-verify-that-dim-ns-a-rank-a-5.26286/invalid@example.com (karush)karush15.3 For the matrix
$$A=\begin{bmatrix}
1 & 0 &0 & 4 &5\\
0 & 1 & 0 & 3 &2\\
0 & 0 & 1 & 3 &2\\
0 & 0 & 0 & 0 &0
\end{bmatrix}$$
(a)find a basis for RS(A) and dim(RS(A)).
ok I am assuming that since this is already in row echelon form, its nonzero rows form a basis for RS(A) then
So...
$$RS(A)=(1,0,0,4,5),\quad(0,1,0,3,2),\quad(0,0,1,3,2)$$
also
dim(RS(A))= ??
(b)verify that dim(NS(A)) + Rank(A) = 5.
ok I am a little unsure what this...
15.3 verify that dim(NS(A)) + Rank(A) = 5]]>0Discrete valuation Ring which is a subring of a field K ProblemFri, 21 Jun 2019 01:02:29 +0000
https://mathhelpboards.com/threads/discrete-valuation-ring-which-is-a-subring-of-a-field-k-problem.26149/
https://mathhelpboards.com/threads/discrete-valuation-ring-which-is-a-subring-of-a-field-k-problem.26149/invalid@example.com (Cbarker1)Cbarker1Dear Everyone,
I am stuck in the middle of a proof. Here is the background information from Dummit and Foote Abstract Algebra 2nd ed.:
Let $K$ be a field. A discrete valuation on $K$ on a function $\nu$: $K^{\times} \to \Bbb{Z}$ satisfying
$\nu(a\cdot b)=\nu(a)+\nu(b)$ [i.e. $\nu$ is a homomorphism from the multiplication group of nonzero elements of $K$ to $\Bbb{Z}$]
$\nu$ is surjective, and
$\nu(x+y)\ge \min{[\nu(x),\nu(y)]}$, for all $x,y\in K^{\times}$ with...
Discrete valuation Ring which is a subring of a field K Problem]]>3Markov process - LimitSat, 15 Jun 2019 18:17:54 +0000
https://mathhelpboards.com/threads/markov-process-limit.26265/
https://mathhelpboards.com/threads/markov-process-limit.26265/invalid@example.com (mathmari)mathmariHey!!
We consider the equation \begin{equation*}u_{k+1}=\begin{pmatrix}a & b \\ 1-a & 1-b\end{pmatrix}u_k \ \text{ with } \ u_0=\begin{pmatrix}1 \\1 \end{pmatrix}\end{equation*}
For which values of $a$ and $b$ is the above equation a Markov process?
Calculate $u_k$ as a function of $a,b$.
Which conditions do $a,b$ have to satisfy so that $u_k$ approximates a finite limit as $k\rightarrow \infty$ and which is that limit?
I have done the following:
So that the equation is a...
Markov process - Limit]]>1Does a line intersect a polygonTue, 11 Jun 2019 17:59:03 +0000
https://mathhelpboards.com/threads/does-a-line-intersect-a-polygon.26244/
https://mathhelpboards.com/threads/does-a-line-intersect-a-polygon.26244/invalid@example.com (sfopeano)sfopeanoHello, I'm wondering if anyone has a formula for determining whether a line intersects a polygon. I would define the line with a starting latitude/longitude and ending latitude/longitude, and I would define the polygon with a series of latitude/longitude coordinates. Many thanks in advance.
-Stephan]]>217.1 Determine if T is a linear transformationMon, 10 Jun 2019 21:16:37 +0000
https://mathhelpboards.com/threads/17-1-determine-if-t-is-a-linear-transformation.26230/
https://mathhelpboards.com/threads/17-1-determine-if-t-is-a-linear-transformation.26230/invalid@example.com (karush)karush17.1 Let $T: \Bbb{R}^2 \to \Bbb{R}^2$ be defined by
$$T \begin{bmatrix}
x\\y
\end{bmatrix}
=
\begin{bmatrix}
2x+y\\x-4y
\end{bmatrix}$$
Determine if $T$ is a linear transformation. So if
$$T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$$
Let $\vec{x}$ and $\vec{y}$ be vectors in $\Bbb{R}^2$ then we can write them as
$$\vec{x}
=\begin{bmatrix}
x_1\\x_2
\end{bmatrix}...
17.1 Determine if T is a linear transformation]]>616.1 Show that e^{2x}, sin(2x) is linearly independent on + infinity -infinityWed, 05 Jun 2019 22:47:17 +0000
https://mathhelpboards.com/threads/16-1-show-that-e-2x-sin-2x-is-linearly-independent-on-infinity-infinity.26224/
https://mathhelpboards.com/threads/16-1-show-that-e-2x-sin-2x-is-linearly-independent-on-infinity-infinity.26224/invalid@example.com (karush)karush16.1 Show that $e^{2x}$, sin(2x) are linearly independent on $(-\infty,+\infty)$
\begin{align*}
w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\
&=??\\
&=??
\end{align*}]]>3Find inner productTue, 04 Jun 2019 23:58:12 +0000
https://mathhelpboards.com/threads/find-inner-product.26218/
https://mathhelpboards.com/threads/find-inner-product.26218/invalid@example.com (Denis99)Denis99Let \(\displaystyle <x, x>=3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} \) be a quadratic form in V=R, where \(\displaystyle x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}\) (in the base \(\displaystyle {e_{1},e_{2},e_{3}}\).
Find the inner product corresponding to this quadratic form.
Is this that easy that you have to change '' second'' x-es for y (for example to write \(\displaystyle 2x_{2}y_{3}\) instead of \(\displaystyle 2x_{2}x_{3}\) at the end), or what I have to do?]]>314.3 Find a basis for NS(A) and dim{NS(A)}Tue, 04 Jun 2019 19:14:32 +0000
https://mathhelpboards.com/threads/14-3-find-a-basis-for-ns-a-and-dim-ns-a.26219/
https://mathhelpboards.com/threads/14-3-find-a-basis-for-ns-a-and-dim-ns-a.26219/invalid@example.com (karush)karushFor the matrix
$A=\left[\begin{array}{rrrrr}
1&0&0&4&5\\
0&1&0&3&2\\
0&0&1&3&2\\
0&0&0&0&0\end{array}\right]$
Find a basis for NS(A) and $\dim{NS(A)}$
$\left[\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5
\end{array}\right]=
\left[\begin{array}{c}
-4x_4-5x_5\\
-3x_4-2x_5\\
-3x_4-2x_5\\
x_4\\
x_5
\end{array}\right]$
ok I just did this but there is duplication in it]]>214.2 find a basis for NS(A) and dim{NS(A)}Tue, 04 Jun 2019 11:58:50 +0000
https://mathhelpboards.com/threads/14-2-find-a-basis-for-ns-a-and-dim-ns-a.26217/
https://mathhelpboards.com/threads/14-2-find-a-basis-for-ns-a-and-dim-ns-a.26217/invalid@example.com (karush)karush$\tiny{370.14.2.}$
For the matrix
$A=\left[
\begin{array}{rrrr}
1&0&1\\0&1&3
\end{array}\right]$
find a basis for NS(A) and $\dim{NS(A)}$
-----------------------------------------------------------
altho it didn't say I assume the notation means Null Space of A
Reducing the augmented matrix for the system $$AX=0$$ to reduced row-echelon form.
$\left[
\begin{array}{ccc}
1 & 0 & 1 \\ 0 & 1 & 3
\end{array} \right]...
14.2 find a basis for NS(A) and dim{NS(A)}]]>1Linear TransformationsTue, 04 Jun 2019 11:54:34 +0000
https://mathhelpboards.com/threads/linear-transformations.26169/
https://mathhelpboards.com/threads/linear-transformations.26169/invalid@example.com (Ereisorhet)EreisorhetGood afternoon people.
So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if:
T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to develop that and find the demonstration.
Thanks for reading.
]]>113.2 verify that ...... is a basis for R^2 find [v]_betaSat, 01 Jun 2019 21:52:16 +0000
https://mathhelpboards.com/threads/13-2-verify-that-is-a-basis-for-r-2-find-v-_beta.26199/
https://mathhelpboards.com/threads/13-2-verify-that-is-a-basis-for-r-2-find-v-_beta.26199/invalid@example.com (karush)karushVerify that
$\beta=\left\{\begin{bmatrix}
0\\2
\end{bmatrix}
,\begin{bmatrix}
3\\1
\end{bmatrix}\right\}$
is a basis for $\Bbb{R}^2$
Then for $v=\left[
\begin{array}{c}6\\8\end{array}
\right]$, find $[v]_\beta$
ok, I presume next is
$c_1\begin{bmatrix}
0\\2
\end{bmatrix}
+c_2\begin{bmatrix}
3\\1
\end{bmatrix}= \left[
\begin{array}{c}6\\8\end{array}
\right]$
by augmented matrix we get (the...
13.2 verify that ...... is a basis for R^2 find [v]_beta]]>214.1 find a vector v that will satisfy the systemSat, 01 Jun 2019 19:58:59 +0000
https://mathhelpboards.com/threads/14-1-find-a-vector-v-that-will-satisfy-the-system.26203/
https://mathhelpboards.com/threads/14-1-find-a-vector-v-that-will-satisfy-the-system.26203/invalid@example.com (karush)karushView attachment 9050
ok I think I got (a) and (b) on just observation
but (c) doesn't look like x,y,z will be intergers so ?????]]>2understanding an example of a basis of R^3Fri, 31 May 2019 19:11:55 +0000
https://mathhelpboards.com/threads/understanding-an-example-of-a-basis-of-r-3.26200/
https://mathhelpboards.com/threads/understanding-an-example-of-a-basis-of-r-3.26200/invalid@example.com (karush)karushView attachment 9046
ok I thot if we get all zeros in 4th column of an augmented matrix that this would be a dependent linear set of vectors
but this example says independent
So I did this
View attachment 9047]]>212.6 linearly dependent or linearly independent?Thu, 30 May 2019 20:03:22 +0000
https://mathhelpboards.com/threads/12-6-linearly-dependent-or-linearly-independent.26198/
https://mathhelpboards.com/threads/12-6-linearly-dependent-or-linearly-independent.26198/invalid@example.com (karush)karushAre the vectors
$$v_1=x^2+1
,\quad v_2=x+2
,\quad v_3=x^2+2x$$
linearly dependent or linearly independent?
if
$$c_1(x^2+1)+c_2(x+2)+c_3(x^2+2x)=0$$
is the system
$$\begin{array}{rrrrr}
&c_1 & &c_3 = &0\\
& &c_2 &2c_3= &0\\
&c_1 &2c_2& = &0
\end{array}$$
I presume at this point observation can be made that this linear dependent
but also...
$$\left[ \begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 1 & 2 & 0 & 0 \end{array} \right]
\sim
\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1...
12.1 - Linear dependence]]>411.1 Determine if the polynominal.... is the span of (...,...)Tue, 28 May 2019 23:14:49 +0000
https://mathhelpboards.com/threads/11-1-determine-if-the-polynominal-is-the-span-of.26186/
https://mathhelpboards.com/threads/11-1-determine-if-the-polynominal-is-the-span-of.26186/invalid@example.com (karush)karushDetermine if the polynomial
$3x^2+2x-1$
is the $\textbf{span}\{x^2+x-1,x^2-x+2,1\}$
ok from examples it looks like we see if there are scalars such that
$c_1(x^2+x-1)+c_2(x^2-x+2,1)=3x^2+2x-1$
so far not sure how this is turned into a simultaneous eq
I did notice that it is common to get over 100 views on these DE problems
so thot it would be good to show sufficient steps]]>410.3 Determine if A is in the span BSat, 25 May 2019 11:34:20 +0000
https://mathhelpboards.com/threads/10-3-determine-if-a-is-in-the-span-b.26162/
https://mathhelpboards.com/threads/10-3-determine-if-a-is-in-the-span-b.26162/invalid@example.com (karush)karushDetermine if $A=\begin{bmatrix}
1\\3\\2
\end{bmatrix}$ is in the span $B=\left\{\begin{bmatrix}
2\\1\\0
\end{bmatrix}
\cdot
\begin{bmatrix}
1\\1\\1
\end{bmatrix}\right\}$
ok I added A and B to this for the OP
but from examples it looks like this can be answered by scalors so if
$c_1\begin{bmatrix}
2\\1\\0
\end{bmatrix}
+
c_2\begin{bmatrix}
1\\1\\1
\end{bmatrix}=\begin{bmatrix}
1\\3\\2
\end{bmatrix}$]]>310.2 Determine if the set of vectors form a vector spaceFri, 24 May 2019 16:53:34 +0000
https://mathhelpboards.com/threads/10-2-determine-if-the-set-of-vectors-form-a-vector-space.26159/
https://mathhelpboards.com/threads/10-2-determine-if-the-set-of-vectors-form-a-vector-space.26159/invalid@example.com (karush)karushDetermine if the set of vectors
$\begin{bmatrix}
x\\y\\5
\end{bmatrix}\in \Bbb{R}^3$
form a vector space
ok if I follow the book example I think this is what is done
$\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix}
+\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix}
+\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix}
=\begin{bmatrix} x_1+x_2+x_3\\y_1+y_2+y_3\\15 \end{bmatrix}$
since the third entry is 15, the set of such vectors is not closed under addition and hence is not a subspace
I assume...
10.2 Determine if the set of vectors form a vector space]]>5Show that S and T are both linear transformationsThu, 23 May 2019 01:06:32 +0000
https://mathhelpboards.com/threads/show-that-s-and-t-are-both-linear-transformations.26136/
https://mathhelpboards.com/threads/show-that-s-and-t-are-both-linear-transformations.26136/invalid@example.com (karush)karush
ok this is a clip from my overleaf hw reviewing
just seeing if I am going in the right direction with this
their was an example to follow but it also was a very different problem
much mahalo]]>7Subspace of three equationsTue, 21 May 2019 15:03:11 +0000
https://mathhelpboards.com/threads/subspace-of-three-equations.26148/
https://mathhelpboards.com/threads/subspace-of-three-equations.26148/invalid@example.com (Johnathon)JohnathonS is the set of solutions for the set of three equations...
x + (1 - a)y^{-1} + 2z + b^{2}w = 0
ax + y - 3z + (a - a^{2})|w| = a^{3} - a
x + (a - b)y + z + 2a^{2}w = b
I worked out...
The first equation is a subset of R^{4} when a = 1, b is any real.
The second equation is a subset of R^{4} when a = 1 or a = 0.
The third equation is a subset of R^{4} when b = 0 and a is any real.
Now, I'm trying to work out the values of a...
Subspace of three equations]]>0Either or statement from Abstract Algebra BookTue, 21 May 2019 12:00:32 +0000
https://mathhelpboards.com/threads/either-or-statement-from-abstract-algebra-book.26140/
https://mathhelpboards.com/threads/either-or-statement-from-abstract-algebra-book.26140/invalid@example.com (Cbarker1)Cbarker1Dear Everyone,
What are the strategies from proving a either-or statements? Is there a way for me to write an either-or statement into a standard if-then statements? For example, this exercise is from Dummit and Foote Abstract Algebra 2nd, "Let $x$ be a nilpotent element of the commutative ring $R$. Prove that $x$ is either a zero or an zero divisor."
I am stuck with an exercise problem. The problem states from Dummit and Foote Ed. 2 Abstract Algebra: "An element $x$ in $R$ (where $R$ is a ring with 1) is called nilpotent if $x^{m}=0$ for some $m \in \Bbb{Z}^{+}$. Show that if $n=a^{k}b$ for some $a,b \in \Bbb{Z}$, then $\overline{ab}$ is a element of $\Bbb{Z}/n\Bbb{Z}$."
My attempt:
Example: When $n=6=3\cdot 2$, then the only element will be $\overline{3*2}$ in $\Bbb{Z}/6\Bbb{Z}$.
Proof: Suppose $n=a^{k}b$...
Nilpotent Problem]]>3Three subspace problemsSat, 18 May 2019 15:29:53 +0000
https://mathhelpboards.com/threads/three-subspace-problems.26134/
https://mathhelpboards.com/threads/three-subspace-problems.26134/invalid@example.com (Johnathon)JohnathonD is the set and the set contains the solutions to
x + (1 - m)y^{-1} + 2z + n^{2}w = 0
I'm trying to find m, n values which means the set is a subspace of R (four dimensions).
===
Similarly, trying to find the m, n values that makes the following two expressions two separate subspaces, too.
mx + y - 3z + (m - m^{2})|w| = m^{3} - m
===
x + (m - n)y + z + 2m^{2}w = n
I've been reviewing the three properties of subspaces over and over and don't...