What is Linear algebra: Definition and 999 Discussions
Linear algebra is the branch of mathematics concerning linear equations such as:
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a
n
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=
b
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{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
linear maps such as:
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,
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↦
a
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{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}
and their representations in vector spaces and through matrices.Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.
Hi
I have noticed that while I have the grasp of the theoretical underpinnings of linear algebra, I need work on applying it to geometric problems (think computer vision and rigid body motion). So, I am looking for a book that allows me to practice 3D geometry problems.
Is there any obvious...
I know for a group to be abelian a*b=b*a
I tried multiplying the matrix by itself also but I’m not sure what I’m looking for.
picture is below of the matrix
https://www.physicsforums.com/attachments/255812
Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A).
Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?
(I can do this with numbers but the trig is throwing me).
I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##.
This can be represented by a diagram:
Thus note...
My idea was to write out the formulas for the orthogonal q vectors in terms of the input vectors using the basics of gram-schmidt. Then, I would rewrite those equations suhc that the a vectors were written in terms of the q vectors. And then, try to find some matrix which would capture the...
Hello, everyone. :)
All I could gather is that, if I'm correct, lattices are spans of the column vectors of the matrix within the "LAT()" notation and the X and Y occurrences are unit placeholders (such as the pixel unit (since this is in the context of image processing)).
And, as an attempt...
Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged.
Hello!
I am struggling with understanding the meaning of "each member is a unit vector":
I can see that N would represent the number of samples, and the pointy bracket represents an...
I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...
Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v =
\begin{pmatrix}
1 \\
1 \\
1
\end{pmatrix},
\begin{pmatrix}
1 \\
0 \\
1
\end{pmatrix},
\begin{pmatrix}
1 \\
1 \\
0
\end{pmatrix}
##.
Not sure what to do next. Thanks!
Has anyone taken these two courses online in a self-paced course for credit? If so, where and how was it in terms of quality? How about price? Opinions/thoughts are much appreciated. I'm working and the closest community college is a commute away, so that's out. I'm finding $1100-3000~ for...
Let's say you were proctoring some test that required proofs of Jordan canonical forms and rational canonical forms.
Would you dock points from a lazy student abbreviating the former as "J-canonical forms" and the latter as "##\mathbb{Q}##-canonical forms" in their proofs?
I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...
I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25.
How do I derive the given equations?
A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 with respect to the basis 𝐵 is upper triangular.
In the proof, he defines U=range(T-𝜆I) (as we have...
Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
I need help to know if I'm on the right track:
Prove/Disprove the following:
Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0.
(V is a vector-space)
I think I need to disprove by using v = 0, however I'm not sure.
Summary: I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space.
I need to do a linear model identification using least squared method.
My model to identify is a...
Here is the initial matrix M:
M = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \end{bmatrix}
I have used the shortcut method outlined in this youtube video: LU Decomposition Shortcut Method.
Here are the row reductions that I went through in order to get my U matrix:
1. R_3 -...
Homework Statement
Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below.
[/B]
Homework Equations
Unsure of relevant equations
The Attempt at...
Homework Statement
Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set.
Homework EquationsThe Attempt at a Solution...
Hello everybody!
I was studying the Glashow-Weinberg-Salam theory and I have found this relation:
$$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...
Let A be a n x n matrix with complex elements. Prove that the a(k) array, with k ∈ ℕ, where a(k) = rank(A^(k + 1)) - rank(A^k), is monotonically increasing.
Thank you! :)
I have already taken two elementary linear algebra courses, and have taken the upper-division linear algebra course offered at my school. However, I feel that I did not learn as much from the latter as I should have. I can owe this to not applying myself as much as I should have, due to other...
So I've taken this Linear Algebra class as an elective. So there's stuff that is so obvious and logically/analytically easy to prove but I honestly don't understand how to prove them using the standard way. So what should I do about this ?
And I really like linear algebra so I don't want to mess...
Homework Statement
Consider the below vector x, which you can copy and paste directly into Matlab. The vector contains the final grades for each student in a large linear algebra course.
x = [59 70 83 89 72 70 54 55 68 61 61 58 75 54 65 55 62 39 43 53 67 100 60 100 61 100 77 60 69 91 82 71 72...
I'm reading about the LU decomposition on this page and cannot understand one of the final steps in the proof to the following:
----------------
Let ##A## be a ##K\times K## matrix. Then, there exists a permutation matrix ##P## such that ##PA## has an LU decomposition: $$PA=LU$$ where ##L## is a...
Homework Statement
Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself,
and the lines through the origin. (Hint: Show that if W is a subspace of
##R^2## that contains two nonzero vectors lying along different lines through
the origin, then W must be all of...
Homework Statement
Let W be a subspace of a vector space V, let y be in V and define the set y + W = \{x \in V | x = y +w, \text{for some } w \in W\} Show that y + W is a subspace of V iff y \in W.
Homework Equations
The Attempt at a Solution
Let W be a subspace of a vector space V, let y...
I was wondering how to measure the first or even the second qubit in a quantum computing system after for example a Hadamard Gate is applied to the system of these qubits: A|00>+B|01>+C|10>+D|11>?
A mathematical and intuitive explanation would be nice, I am a undergraduate sophomore student...
Homework Statement Given the following quadric surfaces:
1. Classify the quadric surface.
2. Find its reduced equation.
3. Find the equation of the axes on which it takes its reduced form.
Homework Equations
The quadric surfaces are:
(1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0 ##...
Hello, I've been working through some Digital Signal Processing stuff by myself online, and I saw a system that I wanted to write down as a Linear Algebra Equation. It's a simple delay feedback loop, looks like this:
The (+) is an adder that adds 2 signals together, so the signal from x[n]...
Hey, I am currently reading over the linear algebra section of the "introduction to quantum mechanics" by Griffiths, in the Inner product he notes: "The inner product of two vector can be written very neatly in terms of their components: <a|B>=a1* B1 + a2* B ... " He also took upon the...
I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc.
But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc.
I am trying to understand some of the subtleties of moving to 3D. In particular, for any...
Homework Statement
Not for homework, but just for understanding.
So we know that if a matrix (M) is orthogonal, then its transpose is its inverse.
Using that knowledge for a diagonalised matrix (with eigenvalues), its column vectors are all mutually orthogonal and thus you would assume that...
Homework Statement
Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...
Homework Statement
In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list.
It's quite long :nb), hope you guys read through it. Thanks! :smile:
Homework Equations
N/A
The Attempt at a Solution...
Homework Statement
Let ##V = \mathbb{R}^4##. Consider the following subspaces:
##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]##
And let ##V = M_n(\mathbb{k})##. Consider the following subspaces:
##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}##
##V_2 =...
(a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?
Homework Statement
The SO(3) representation can be represented as ##3\times 3## matrices with the following form:
$$J_1=\frac{1}{\sqrt{2}}\left(\matrix{0&1&0\\1&0&1\\ 0&1&0}\right) \ \ ; \ \ J_2=\frac{1}{\sqrt{2}}\left(\matrix{0&-i&0\\i&0&-i\\ 0&i&0}\right) \ \ ; \ \...
I have a question about HHL algorithm https://arxiv.org/pdf/0811.3171.pdf for solving linear equations of the form:
A x = b
Where A, x and b are matrices
Take for example
4x1 + 2x2 =14
5x1 + 3x2 = 19
HHL apply the momentum operator eiAτto/T on the state, do a Fourier Transform on |b> and...
I'm a physics student who has the option to take some advanced math courses (Real analysis through Rudin and beyond, functional analysis if I have time, as well as algebra through Artin). I'm only just going into my second year this term, and will either be retaking linear algebra 2, or taking...
Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as
$$p \otimes q := \begin{pmatrix}
p(1)q(1) \\
p(1)q(2) \\
\vdots \\
p(1)q(n) \\
\vdots \\...
Hello. I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20. It states that:
Let A be an n by n matrix. Let h:R^n->R^n be the linear transformation h(x)=A x. Let S be a rectifiable set (the boundary of S BdS has measure 0) in R^n. Then v(h(S))=|detA|v(S)...
Homework Statement
Suppose that AB = 0, where A is a 3 x 7 full rank matrix and B is 7 x 53. What is the highest possible rank of matrix B.
Homework EquationsThe Attempt at a Solution
Since each column of B is in the null space of A, the rank of B is at most 4.
I don't understand why it is...
Hello,
I'm having a visualisation problem. I have a point in R3 that I want to rotate about the ##y##-axis anticlockwise (assuming a right-handed cartesian coordinate system.) I know that the change to the point's ##x## and ##z## coordinates can be described as follows:
$$z =...
Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of...