1. having two distinct real eigenvalues,

$A=\begin{bmatrix}

2 & -1\\

-1& 2

\end{bmatrix},\quad \left| \begin{array}{rr} 2 - \lambda & -1 \\ -1 & 2 - \lambda \end{array} \right|=\lambda^{2} - 4 \lambda + 3 \therefore \lambda_1=3\ \lambda_2=1$

2. a pair of complex eigenvalues.

$\left| \begin{array}{cc}

2 - \lambda & 1 \\

-1 & 2 - \lambda

\end{array} \right|=\lambda^{2} - 4 \lambda + 5\quad \lambda_{1}=2 - i,\lambda_{1}=2 + i$

3. two...

307 Construct 3 different 2x2 matrices]]>

A. –6

B. –4

C. 4

D. 6

E. 8

Because the curve's symmetrical axis is x = a, then:

\(\displaystyle -\frac{2p}{2(1)}=a\)

–p = a

a – f(a) = –p + (–p) = 0

I got zero. Is there anything I did wrong?]]>

A zoo has 80 cotton-top tamarins. Describe in detail how the random- number table given below could be used to select a sample of 5 of them, for a study of tail lengths.

8330 3992 1840 0330 1290 3237 9165 4815 0766

(5marks)

So im not really sure where to go with this one, i can't see it being any type of systematic so it must be simple random sampling, but the issue is, there are nine groups of four numbers and I can't see a way to get these 36 numbers to...

Random Sampling Question]]>

v=\left[\begin{array}{r}

-3\\-4\\-5\\4\\-1

\end{array}\right]

w=\left[\begin{array}{r}

-2\\0 \\1 \\4 \\-1

\end{array}\right]

x=\left[\begin{array}{r}

2\\3 \\4 \\-5 \\0

\end{array}\right]

y=\left[\begin{array}{r}

-2\\1 \\0 \\-2 \\7

\end{array}\right]

z=\left[\begin{array}{r}

-1\\0 \\2 \\-3 \\5

\end{array}\right]

$

Construct matrices not yet row reduced echelon form whose null space consists all linear combinations of

1. just x

2. just y

3. just z

ok I presume this...

matrices......whose null space consists all linear combinations]]>

A. –8

B. –5

C. –2

D. 2

E. 8

Since the equation has positive roots then \(\displaystyle x_1>0\) and \(\displaystyle x_2>0\) thus \(\displaystyle x_1+x_2>0\) and \(\displaystyle x_1x_2>0\)

\(\displaystyle x_1+x_2>0\)

\(\displaystyle \frac{-(-4a)}{a-1}>0\)

\(\displaystyle x_1x_2>0\)

\(\displaystyle \frac{4a+7}{a-1}>0\)

However I progressed, I couldn't determine a as a single value and...

[ASK] Stuck on a Quadratic Equation]]>

I wanted to ask you if you are familiar with html, css, javascript and php... What is it about?

Here is for example an exercise:

Implement in HTML and CSS the site that is shown below (the specifications are marked with red). The site will be without functionality. How would we proceed? ]]>

Please proceed with the following calculations, based on the data you can find in the attached Excel sheet:

- Forecast the
**ARPU**s for: Determine the ARPU discounts that are needed by country in order to deliver*voice services by country***cumulative reduction**of 27% by the end of year 3 and 35% by the end of year 4 - Forecast the
**Actual savings of the customer**over the life of the dealand*in total**by...*

Case Study for Commercial Delivery Specialist position]]>

]]>

a) Find an equation of in the form d=mt+b

b) Determine the slope and d intercept and explain what they mean

c) How far will John be from the sensor 5s after he begins walking?]]>

We have a triangle $ABC$. A point $E$ is on side $c$ such that $\overline{AE}=\frac{1}{5}\cdot \overline{AB}$, a point $F$ is on side $b$ such that $\overline{FC}=\frac{1}{4}\cdot \overline{AC}$. The segments $EC$ and $FB$ intersect on $S$. Write $BS$ in relation to $BF$ and $CS$ in relation to $CE$.

Can we solve that using vectors? ]]>

Write the slope-intercept equation of the line that is parallel to -9x-7y=4 and has the same y-intercept as the graph of -5x+11y=-22.]]>

An insurance office has $2500$ contracts with mean annual profit (per contract) $\mu=330$ and standard deviation $\sigma=540$. Calculate the probability that the total annual profit is not more than $800000$.

I have one the following:

The annual total profit should be not more than $800000$, that means that per contract it shoulebe not more than $\frac{800000}{2500}=320$.

So $$Z=\frac{X-\mu}{\sigma}=\frac{320-330}{540}\approx -1.31 \\ P(Z\leq -1.31)=0.0968$$ Is that the correct...

Probability that the total annual profit is not more than 800000]]>

Some people assume that a specific car model does at least $\mu_0=120$ km with $1$ Lt benzin.

$10$ independent tests give the following results: $$104, \ 96, \ 80, \ 100, \ 108, \ 100, \ 112, \ 120, \ 130, \ 132$$

(a) Give the Null Hypothesis $H_0$ and the alternative Hypothesis $H_1$, for the test of that assumption.

(b) Give the statistic function of that test.

(c) late the p-value of the test.

(d) In what confidence level can the assumption be rejected?

For (a) is...

Hypothesis Test]]>

I am having trouble solving part 2, for

$ W_{\frac{n(n+1)}{2}} \leq 2^{n} (n-1) + 1 , n \geq 0 $

I know that $W_{m} \leq 2*W_{m-k} + 2^{k} – 1, 0 \leq k \leq m$

Let $m = \frac{n(n+1)}{2}$

So now $W_{\frac{n(n+1)}{2}} \leq 2*W_{\frac{n(n+1)}{2} - k} + 2^{k} - 1, 0 \leq k \leq \frac{n(n+1)}{2}$

Let k = n (just a hunch, I can't really explain why except that because the equation has an n inside)

$W_{\frac{n(n+1)}{2}} \leq 2*W_{\frac{n(n-1)}{2}} + 2^{n} –...

4 rod Tower of Hanoi proof]]>

Show that the plane H defined by:

$H=\left\{

\alpha_1\left[

\begin{array}{rrr}1\\1\\1\end{array} \right]

+\alpha_2\left[\begin{array}{rrr}1\\-1\\0\end{array} \right]

\textit{ Such that } \alpha_1,\ \alpha_1\in\mathbb{R}\right\}

=\begin{bmatrix}a_1+a_2\\ a_1+a_2\\ a_1\end{bmatrix}$

$\text{rref}(H)=\left[ \begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & 0 \end{array} \right]$

ok I don't know what this answers]]>

ok I didn't understand how they got the eigenspaces

the original matrix was

$A=\left[\begin{array}{rrr}−1&2\\−6&6\end{array} \right]$

so think I got values correct $\lambda=2,3$

307 hw]]>

I am a philologist who is fond of mathematics, but who unfortunately has just an elementary high school knowledge of them. I am translating

Thank you very much in...

An application of the law of cosines?]]>

Co basically I've got this table in excel, where X (row) is a width and Y is a height (column) of a wooden sauna cabin, the X;Y is the price for a sauna with said dimensions. I need to find a relationship between the size of the sauna and...

Need help with a work related math problem]]>

-----

If $x,\,y$ and $z$ are real numbers satisfying

$(x+1)(y+1)(z+1)=3\\(x+2)(y+2)(z+2)=-2\\(x+3)(y+3)(z+3)=-1$

find the value of $(x+20)(y+20)(z+20)$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

-----

Find all integer solutions of the system

$xz-2yt=3\\

xt+yz=1$

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

N = any natural number

1 = xUx

2 = xUxUx

What exactly is the difference between 1 and 2? Is each instance of x different? If so, how?]]>

definition

ok I think this is the the definition we need for this practice exam question,

However I tried to insert using a link but not successful

I thot if we use a link the image would always be there...

Q01 are linearly independent vectors, so are....]]>

Let $g(x)-=x-x^3$. The point $x=0$ is a fixed point for $g$. Show that if $x^{\star}$ is a fixed point of $g$, $g(x^{\star})=x^{\star}$, then $x^{\star}=0$. If $(x_k)$ the sequence $x_{k+1}=g(x_k)$, $k=0,1,2,\ldots$ show that if $0>x_0>-1$ then $(x_k)$ is increasing and converges to $0$.

$g(x)=x-x^3$

$g(0)=0$

$g(x^{\star})=x^{\star} \Rightarrow x^{\star} -{x^{\star}}^3=x^{\star} \Rightarrow {x^{\star}}^3=0 \Rightarrow...

Fixed point,, Jacobi- & Newton Method, Linear Systems]]>

I’m looking for assistance in determining and calculating the size of a lottery draw pattern matrix, using simple mathematics formulas. That I myself can learn to use to include math formulas on the information pages of my new Lotto Probability Draw Pattern Mathematics Database Probability Reports, that are located on my website @ the Tinkermen Lotto Report here: Draw...

Mathematically determine a Lotto Draw Pattern Matrix]]>

If $f\in O(\Delta (0,1,15))$ then does it hold that $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=2\pi i\text{Res}\left (\frac{f(z)}{(z-6+4i)^5}, 6-4i\right )+\int_{C(0,6)}\frac{f(z)}{(z-6+4i)^5}\, dz$$ Do we maybe use here Cauchy theorem and then we get $$\int_{C(0,10)}\frac{f(z)}{(z-6+4i)^5}\, dz=0$$

Is there a sequence of holomorphic polynomials $P_n(z), n=1,2,\ldots$ such that $$P_n(z)\rightarrow...

Complex Analysis]]>

Provide an interpretation of this value.

$\begin{array}{rrrr}

x & y \\

12.17 & 1.88 \\

11.70 & 1.82 \\

11.63 & 1.77 \\

12.27 & 1.93 \\

12,03 & 1.83 \\

11.60 & 1.77 \\

12.15 & 1.83 \\

11.72 & 1.83 \\

11.30 & 1.70

\end{array}$

here is my desmos plot and I can see that R^2 is $83.0\%$

but after looking at some examples I don't see how it is derived

However, the interpretation of this is

of the variability in y is explained by the least-squares...

d. The coefficient of determination is 83.0 %]]>

Let $N_j$, $j=-k,\ldots , m-1$ the normalized B-splines of the set of nodes $x_0, \ldots , x_m$ of degree $k$.

Show that $$\sum_{j=-k}^{m-1}N_j(x)=1 \ \text{ for all } x\in [x_0, x_m]$$

A formula with divided differences is

\begin{align*}&N_j(x)=(x_{j+k+1}-x_j)B_j(x) \\& \text{ with } \ B_j=(\cdot -x)_+^k[x_jx_{j+1}x_{j+2}\ldots x_{j+k+1}] =\frac{(\cdot -x)_+^k[x_{j+1}x_{j+2}\ldots x_{j+k+1}]-(\cdot -x)_+^k[x_jx_{j+1}x_{j+2}\ldots x_{j+k}]}{x_{j+k+1}-x_j}\end{align*}...

Sum of normalized B-splines]]>

We have the following linear maps \begin{align*}\phi_1:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto \begin{pmatrix}x+y\\ x-y\end{pmatrix} \\ \phi_2:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto \begin{pmatrix}-y\\ x\end{pmatrix} \\ \phi_3:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto \begin{pmatrix}y\\ 0\end{pmatrix} \end{align*}

1. Give (if possible) for each $i\in...

Give a basis to get the specific matrix M]]>

One of the techniques we are using at the digital communications to improve the reliability of a noisy communication channel, is to repeat a symbol many times.

For example, we can send each symbol $0$ or $1$ say three times. More precisely, applying the rule of majority, a $0$ (or $1$) is sent as $000$ (or $111$ respectively) and is decoded at the receiver with $0$ (or $1$) if and only if the received sequence of three symbols contains at least two $0$ (or $1$...

Probability to get the correct message]]>

A cyclist, constantly driving forward, covers a distance of $80$ km in exactly $2$ hours. Show that there is a $1$-kilometer section of this journey that he covers in exactly $1.5$ minutes.

Hint: Continuity of the inverse function.

So we have to define a function $\phi :T\rightarrow K, \ x\mapsto \frac{120x}{80}$, where $T$ is the set of time and $K$ is the set of kilometer. Or can we not just define the function like that?

Then we want to show that there is a $y$ such...

Show that there is a 1-km section that he covers in 1.5 minutes]]>

An aeroplane flies over a tower of height $h> 0$ at height $H> h$. At what distance $x$ is the angle $\alpha$ at which the tower is seen from the aeroplane, maximum?

(You can use elementary geometry and that $\arctan'(x)=\frac{1}{1+x^2}$.)

From Pythagorean Theorem for the larger triangle we have that $H^2+x^2=:y^2$.

Do we apply for the smaller triangle law of cosine? But the upper side of that triangle is not known,and not related...

When is the angle maximum?]]>

Let $V$ be a $\mathbb{R}$-vector space, let $x,y\in V$ and let $U,W\leq_{\mathbb{R}}V$ be subspaces of $V$.

Show that :

(a) If $(x+U)\cap (y+W)\neq \emptyset$ and $z\in (x+U)\cap (y+W)$ then $(x+U)\cap (y+W)=z+(U\cap W)$.

(b) The following statements are equivalent:

(i) $U=W$ and $x-y\in U$.

(ii) $x+U=y+W$.

I have done the following :

(a) let $z\in (x+U)\cap (y+W)$. That means that $z\in x+U$ and $z\in y+W$. So we have that $z=x+u$, for $u\in U$ and $z=y+w$, for $w\in W$...

Statements about subspaces]]>

Let \begin{equation*}A=\begin{pmatrix}0 & -2 & 2 & 0 & 0 & -6 \\ 0 & 0 & 0 & 1 & 1 & 3\\ 0 & 0 & 0 & 1 & -1 & -1 \\ 0 & 1 & -1 & 0 & 2 & 7\\ 0 & 3 & -3 & 1 & 2 & 14\end{pmatrix}, \ b_1=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0\end{pmatrix} , \ b_2=\begin{pmatrix}-2 \\ 1 \\ -1 \\ 3 \\ 5\end{pmatrix}, \ b_3=\begin{pmatrix}-2 \\ 2 \\ 0 \\ 4 \\ 7\end{pmatrix}\end{equation*}

(a) Determine the row echelon form of $A$.

(b) Calculate for all $1\leq i\leq 3$ with $L(A,b_i)\neq...

Row echelon form : Can the first column contain only zeros?]]>

I want to prove the following:

If $x_0$ is an inner point of $D$ ($x_0 \in \text{int } D$), so the differentiability of $f$ at $x_0$ is equivalent to each of the following two conditions.

(i) $\exists \alpha\in \mathbb{C}$ : $\forall \epsilon>0 \ \exists \delta>0\ \forall x\in B(x_0,\delta): \ |f(x)-f(x_0)-\alpha(x-x_0)|\leq \epsilon |x-x_0|$

(ii) There is a $\delta>0$, $\alpha\in \mathbb{C}$ and $r\in B(x_0, \delta)\rightarrow \mathbb{C}$ continuous at $x_0$, $r(x_0)=0$...

Differentiability of f]]>

Make a drawing for each of the values of the angle below indicating the angle at the unit circle (in other words: $\text{exp} (i \phi )$) and its sine, show cosine, tangent and cotangent.

Give these four values explicitly in every case (you are allowed to use elementary geometry and the Pythagorean theorem).

$$\phi=\frac{\pi}{6}, \ \ \phi=\frac{\pi}{4}, \ \ \phi=\frac{2\pi}{3}, \ \ \phi=\frac{5\pi}{6}, \ \ \phi=-\frac{2\pi}{3}, \ \ \phi=-\frac{\pi}{3}$$

So at a unit...

Angles in unit circle]]>

Let $\lambda\in \mathbb{R}$ and \begin{equation*}a=\begin{pmatrix}1 & 2 &-1& \lambda & -\lambda \\ 0 & 1 & -1& \lambda & 2\\ 2 & 2 & 1 & 1 & 3\lambda-1 \\ 1 & 1 & 1 & \lambda & 5\end{pmatrix}\in \mathbb{R}^{4\times 5}\end{equation*}

(a) Let $\lambda=1$. Determine a Basis $\mathcal{B}$ of $\mathbb{R}^5$ and a Basis $\mathcal{C}$ of $\mathbb{R}^4$, such that $\mathcal{M}_{\mathcal{C}}^{\mathcal{B}}(\phi_a)$ at the upper left side there is an unit matrix and elsewise zero.

(b)...

M has the unit matrix at the upper left side and zero everywhere else]]>

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The numbers $x_1,\,x_2,\,\cdots,\,x_{1991}$ satisfy the equation $|x_1-x_2|+|x_2-x_3|+\cdots+|x_{1990}-x_{1991}|=1991$.

What is the greatest possible value of the expression $|y_1-y_2|+|y_2-y_3|+\cdots+|y_{1990}-y_{1991}|$, where $y_k=\dfrac{1}{k}(x_1+x_2+\cdots+x_k)$?

-----

Remember to read the POTW submission guidelines to find out how to...

Problem Of The Week #454 February 8th 2021]]>

Let $X_1, \ldots , X_n$ be independent, identically distributed random variables with $$P(X_i=-1)=P(X_i=1)=\frac{1}{2}$$

We consider the random variables $Y_i=\max \{X_i,X_{i+1}\}$, $i=1,\ldots , n-1$.

(a) Determine the distribution of $Y_i$, $i=1,\ldots , n-1$.

(b) Calculate the expected value of $Y_i$, $i=1,\ldots , n-1$.

(c) Calculate the covariance of $Y_i$ and $Y_j$, i.e. $\text{Cov}(Y_i, Y_j)=E(Y_iY_j)-E(Y_i)E(Y_j)$, $i,j=1,\ldots , n-1$.

For (a) we have :

The...

Calculate the covariance]]>

$\qquad\textit{mean}=\dfrac{\textit{sum}}{\textit{data set}}=\dfrac{48}{8}=\textbf{6}$

The variance of this data set is

$\qquad \textit{new mean} =\textit{current mean}\cdot \textit{scalar}=6\cdot 3=\textbf{18}$

OK I didn't know how to get...

b. Find the value of the new variance]]>

]]>

Let $X$, $Y$ and $Z$ be independent random variables. Let $X$ be Bernoulli distributed on $\{0,1\}$ with success parameter $p_0$ and let $Y$ be Poisson distributed with parameter $\lambda$ and let $Z$ be Poisson distributed with parameter $\mu$.

(a) Calculate the distribution, the expected value and the variance of $XY$.

(b) Determine the Covariance and the correlation between $XY$ and $XZ$.

For question (a) :

We have that $$P(X=0)=1-p_0 \ \text{ and} \...

Distribution, expected value, variance, covariance and correlation]]>

i) How tall must the tree be to stop the...

Quadratic Relation]]>

We've updated the TikZ picture functionality.

For those of you who are new to the site, this previous announcement describes what the TikZ functionality is about.

It also serves as reference for current and future functionality.

What's new:

- The pictures are transported over HTTPS now to make them secure.

This is in particular important for Google Chrome, which has tightened its security.

- The...

TikZ pictures update]]>

Thanks]]>

I am a philologist who is fond of mathematics, but who unfortunately has just an elementary high school knowledge of them. I am translating

Demonstration of a formula for the ratio between the hypotenuses of two triangles]]>

\[ \sum_{n=0}^{N}\frac{a}{2^{n}}\sin^{2}\left(\frac{a}{2^{n}}\right) \]

If we plot or evaluate the value then it does appear that the series converges very fast...

Find limit]]>

x^5-15x^3-x-60 = 0

How do I get started? I think the solution is not over the real numbers.

You say?]]>

I want to calculate the derivatives of the below functions.

1. $\displaystyle{f(x)=x^n\cdot a^x}$, $\in \mathbb{N}_0, x\in \mathbb{R},a>0$

2. $\displaystyle{f(x)=\log \left [\sqrt{1+\cos^2(x)}\right ]}$,$x\in \mathbb{R}$

3. $\displaystyle{f(x)=\sqrt{e^{\sin \sqrt{x}}}}$, $x>0$

4. $\displaystyle{f(x)=x^p}$, $x>0, p\in \mathbb{R}$

5. $\displaystyle{f(x)=\left (1-\sqrt{2}\sin \left (\frac{x}{2}\right )\right )\cdot \sqrt{1+\tan^2(x)}\cdot \left (1+\sqrt{2}\sin \left...

Calculate derivatives]]>

Let $f(x)=e^{-x}\sin (x)$, $x\in \mathbb{R}$.

a) Calculate the Taylor polynomial of order $4$ at $0$.

b) Calculate the value of $f \left (\frac{1}{2}\right )$ using estimation for the remainder with an error not more than $\frac{1}{400}$.

I have done question a) :

\begin{align*}T_{0,4}(x)&=\sum_{k=0}^4\frac{f^{(k)}(0)(x-0)^k}{k!}=\frac{f^{(0)}(0)x^0}{0!}+\frac{f^{(1)}(0)x^1}{1!}+\frac{f^{(2)}(0)x^2}{2!}+\frac{f^{(3)}(0)x^3}{3!}+\frac{f^{(4)}(0)x^4}{4!} \\ & =\frac{0\cdot...

Value of f(1/2) using estimation for the remainder]]>

find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$

$S=\left\{

\left[\begin{array}{r}2\\-1\\2\end{array}\right],

\left[\begin{array}{r}3\\0\\1\end{array}\right],

\left[\begin{array}{r}1\\1\\-1\end{array}\right],

\left[\begin{array}{r}5\\-1\\3\end{array}\right]

\right\}$

make matrix A and derive RREF(A) to find pivot columns

$A=\left[

\begin{array}{rrrr}

2 & 3 & 1 & 5 \\

-1 & 0 & 1 & -1 \\

2 & 1 & -1 & 3

\end{array} \right]

\quad...

115.C51 find a linearly independent set T so that T=S]]>

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Find all integer solutions of the equation $\left\lfloor \dfrac{x}{1!} \right\rfloor+\left\lfloor \dfrac{x}{2!} \right\rfloor+\cdots+\left\lfloor \dfrac{x}{10!} \right\rfloor=1001$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

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Determine all positive integers $a,\,b$ and $c$ that satisfy the following equation:

$(a+b)!=4(b+c)!+18(a+c)!$

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

If $\sum\limits_{i=1}^{50} f(i)=90$ and $\sum\limits_{i=30}^{50} g(i)=60$, what is the value of $\sum\limits_{i=1}^{50} (7 g(i)-f(i)+12)/(2)$?

P.S. To those who could answer this, it would be a great help for me! Thank you so much!]]>

Solution:

Let x = first account

Let y = second account

The words "up to" tells me to use less than or equal to when adding the two accounts.

The words...

Systems of Inequalities]]>

(a) Write a system of inequalities that describes the constraints on the circles.

What does the word CONSTRAINTS mean here?

(b) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in

the context of the problem.

What exactly is part (b) asking for?]]>

Let $C$ be a $\mathbb{R}$-vector space, $1\leq n\in \mathbb{N}$ and let $\phi_1, \ldots , \phi_n:V\rightarrow V$ be linear maps.

I have shown by induction that $\phi_1\circ \ldots \circ \phi_n$ is then also a linear map.

I want to show now by induction that if $V$ is finite then $\text{Rank}(\phi_1\circ \ldots \circ \phi_n)\leq \min \{\text{Rank}(\phi_i)\mid 1\leq i\leq n\}$.

Rank of composition of linear maps]]>

$\dfrac{|a_1+.......a_n|}{1+|a_1+.........+a_n|}\leq\dfrac{|a_1|}{1+|a_1|}+...........\dfrac{|a_n|}{1+|a_n|}$]]>

Use elementary row and column operations to transform \[ \begin{bmatrix} I_{n} & 0 \\ 0 & AB \end{bmatrix} \] to \[ \begin{bmatrix} B & I_{n} \\ 0 & A \end{bmatrix} \].]]>

ok I am trying to solve some other problems following this example but can[t see how the $z_1,z_2,z_3$ are created

I know it is pulled for REFF matrix]]>

]]>

solve for y]]>

]]>

]]>

Let b = birds

Met a bird = (b + 1)

Half of us plus you = (b + 1)/2 + (b + 1)

The equation is:

(b + 1)/2 + (b + 1) = 100

Yes?]]>

I say we take the log on both sides as step one.

Yes?]]>

]]>

I am in the process of relearning high school geometry through Khan Academy. I am current an graduated undergraduate student in mathematics. I am doing this because geometry is one of my weakest subject in mathematics. Second reason is that I want to reason out a problem geometrically. I also want to relearn my university level geometry textbook. I have a hard time with spatial reasoning in general. I am wondering why does learning the rigid transformations and dilations and...

How does rigid transformation and dilation help with learning Geometry?]]>

and not associative.

ok I read all I could on trying to understand this but seem to not see something simple

if we have the example of

$u=\begin{bmatrix}2\\-3\\4\\2\end{bmatrix} v=\begin{bmatrix}-1\\5\\2\\-7\end{bmatrix} u+v=\begin{bmatrix}2\\-3\\4\\2\end{bmatrix}+\begin{bmatrix}-1\\5\\2\\-7\end{bmatrix} =\begin{bmatrix}2+(-1)\\-3+5\\4+2\\2+(-7)\end{bmatrix}... |

T31 vector subtraction is not commutative and not associative.]]>

We have the glide reflection

\begin{equation*}\kappa \begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}x+2\\ -y+2\end{pmatrix}\end{equation*}

and the rotation

\begin{equation*}\delta \begin{pmatrix} x\\ y\end{pmatrix}=\begin{pmatrix} -x +2\\ -y+2\end{pmatrix}\end{equation*}

The composition of these maps is

$$\kappa \left (\delta \begin{pmatrix}x \\ y\end{pmatrix}\right )=\kappa \begin{pmatrix} -x +2\\ -y+2\end{pmatrix}=\begin{pmatrix} -x +4\\ -y+4\end{pmatrix}$$

What kind of...

What kind of transformation is the composition?]]>

20 ≤ x ≤ 70. The American Heart Association recommends that when a person exercises, the person should strive for a heart rate that is at least 50% of the maximum and at most 85% of the maximum.

(a) Write a system of inequalities that describes the exercise target heart rate region.

I need someone to get me started here.

(b) Find two solutions of the system and interpret their...

Heart Rate Formula]]>

My Effort:

Circumference = pi•d

10 •pi = pi•d

10•pi/pi = d

10 = d, where d is the diameter of the circle.

Area = pi•r^2, where r is the radius of the circle.

Diameter = 2 times the radius.

10pi = 2r

10pi/2 = r

5pi = r

A = pi•r^2

A = pi(5pi)^2

A = 25•pi^3, which makes no sense.

Only the volume is cubed. This is not a volume question.]]>