Recent questions in Matrix transformations

Matrix transformations
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Cory Patrick
2022-06-22

Matrix transformations
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Reed Eaton
2022-06-22

Matrix transformations
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Davon Irwin
2022-06-21

We want to compute the matrix representation $[T]$ from $\alpha $ to $\gamma $ coordinates.

${M}_{2\times 2}$ has the standard ordered basis "$\alpha $" for a $2\times 2$ matrix. $F$ has the standard basis "$\gamma $" for a scalar.

My understanding is that any matrix $A$ in the space ${M}_{2\times 2}$ can be represented by the linear combination:

$a{\alpha}_{1}+b{\alpha}_{2}+c{\alpha}_{3}+d{\alpha}_{4}$

and the $tr(A)$ can be written as:

$(a+c)\gamma .$

I'm not sure how to get the matrix representation from this.

Matrix transformations
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Leonel Contreras
2022-06-21

A reflection about the $x$-axis followed by a rotation of $\frac{\pi}{6}$

Matrix transformations
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boloman0z
2022-06-21

Matrix transformations
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Roland Waters
2022-06-20

need to express the transformation matrix, N, from B to C by M. ($\mathrm{\forall}v\in V[v{]}_{C}=N[v{]}_{B}$)

Is this true?

$({w}_{1}\dots {w}_{n}{)}^{-1}({v}_{1}\dots {v}_{n})=N\phantom{\rule{0ex}{0ex}}{\textstyle (}({v}_{1}\dots {v}_{n}){M}^{t}{)}^{-1}({v}_{1}\dots {v}_{n})=N\phantom{\rule{0ex}{0ex}}({M}^{t}{)}^{-1}({v}_{1}\dots {v}_{n}{)}^{-1}({v}_{1}\dots {v}_{n})=N\phantom{\rule{0ex}{0ex}}({M}^{-1}{)}^{t}=N$

Matrix transformations
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Dale Tate
2022-06-20

$\left[\begin{array}{cccccc}a& b& c& d& e& f\end{array}\right]$

which I would like to convert into a 2D matrix. Let's arbitrarily say I'd like to ravel along the rows first (fill one row before moving to the next).

$\left[\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right]$

Is there a series of multiplicative matrix transformations that performs this reshaping, and if so, what is the general name for this operation?

Matrix transformations
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hawatajwizp
2022-06-20

I took the standard basis for grade-n polynomials:

$B=<1,x,{x}^{2},\dots ,{x}^{n}>,\phi (B)=\phi (1,x,{x}^{2},...,{x}^{n})=(0,2x,...,n{x}^{n-1})$

So, the matrix is $\left[\begin{array}{c}0\\ 2x\\ \dots \\ n{x}^{n-1}\end{array}\right]$?

Matrix transformations
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watch5826c
2022-06-20

For the following 3D transfromation matrix M, find its inverse. Note that M is a composite matrix built from fundamental geometric affine transformations only. Show the initial transformation sequence of M, invert it, and write down the final inverted matrix of M.

$M=\left(\begin{array}{cccc}0& 0& 1& 5\\ 0& 3& 0& 3\\ -1& 0& 0& 2\\ 0& 0& 0& 1\end{array}\right)$

Matrix transformations
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vittorecostao1
2022-06-19

Matrix transformations
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Feinsn
2022-06-17

$T\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 5\\ 6\end{array}\right)\text{}\text{and}\text{}T\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)=\left(\begin{array}{c}5\\ 6\\ 7\end{array}\right).$The question told me to find $T\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)$ which I did but I am wondering how would I find the transformation matrix $T$?

Matrix transformations
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Quintin Stafford
2022-06-17

The Matrix R is below:

$\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$

And i have a Linear Transformation below here below that defines by this below.

The linear Transformation $S:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ is given by $S(x)=R(x)+3x.$

how to get the Matrix $S$ from the definition above.

Matrix transformations
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glycleWogry
2022-06-16

Given two non-zero vectors $a=({a}_{1},{a}_{2})$ and $b=({b}_{1},{b}_{2})$ we define $\sim (a,b)$ by $\sim (a,b)={a}_{1}{b}_{2}-{a}_{2}{b}_{1}$. Let $A$, $B$ and $C$ be points with position vectors a, b and c respectively, none of which are parallel. Let $P$, $Q$ and $R$ be points with position vectors $p$, $q$ and $r$ respectively, none of which are parallel.

(i) Show that there exist $2\times 2$ matrix $M$ such that $P$ and $Q$ are the images of $A$ and $B$ under transformation represented by $M$.

Matrix transformations
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George Bray
2022-06-16

${x}_{1}={X}_{1}+2\lambda {X}_{2}$

${x}_{2}={X}_{2}-\lambda {X}_{1}$

is:

$\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{cc}1& 2\lambda \\ -\lambda & 1\end{array}\right]\left[\begin{array}{c}{X}_{1}\\ {X}_{2}\end{array}\right]$

but what if I have the non-linear transformation:

${x}_{1}={X}_{1}+2\lambda {X}_{2}{X}_{1}^{2}$

${x}_{2}={X}_{2}-\lambda {X}_{1}$

How can it be expressed in matrix form to get it's inverse?

Matrix transformations
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April Bush
2022-06-16

Matrix transformations
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enrotlavaec
2022-06-16

Let $A=\left[\begin{array}{cc}5& -3\\ 2& -2\end{array}\right]$, which is a linear transformation ${\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$. Find the matrix representing the transformation with respect to basis $\left[\begin{array}{cc}3& 1\\ 1& 2\end{array}\right]$.

I understand how to solve the question. You would do ${B}^{-1}AB$. I did the multiplication and got $\left[\begin{array}{cc}20& 0\\ 0& -5\end{array}\right]$. The author of the document got the same answer but then multiplied the matrix by 1/5 to get $\left[\begin{array}{cc}4& 0\\ 0& -1\end{array}\right]$. Why is he allowed to do that?

Matrix transformations
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Misael Matthews
2022-06-16

Matrix transformations
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Dwllane4
2022-06-16

$\left[\begin{array}{ccc}x{x}^{\prime}& x{y}^{\prime}& x{z}^{\prime}\\ y{x}^{\prime}& y{y}^{\prime}& y{z}^{\prime}\\ z{x}^{\prime}& z{y}^{\prime}& z{z}^{\prime}\end{array}\right]$

But shouldn't it be the transpose of matrix given above as the transformation matrix is given by coordinates of transformations of bases ?

Matrix transformations
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excluderho
2022-06-15

Matrix transformations
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Santino Bautista
2022-06-15

(1) For $\theta \u03f5\mathbb{R}$ fixed, $R:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{2}},R(x,y)=(x\mathrm{cos}(\theta )-y\mathrm{sin}(\theta ),x\mathrm{sin}(\theta )+y\mathrm{cos}(\theta ))$ is linear.

(2) The transformation $T:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{2}},T(x,y)=(1+{x}^{2},1+{y}^{2})$ is linear.

(3) For $n\ge 1$, every linear transformation $L:R:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ satisties $L(\mathbf{0})=\mathbf{0}$

(4) If $L:{\mathbb{R}}^{\mathbb{3}}\to {\mathbb{R}}^{\mathbb{2}}$ $L:{\mathbb{R}}^{\mathbb{3}}\to {\mathbb{R}}^{\mathbb{2}}$ is linear, then the matrix which represents $L$ with respect to the standard coordinates must have positive nullity.

(5) If $L:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{3}}$ is linear, then the matrix which represents $L$ with respect to the standard coordinates must have positive nullity.

(A) Only (1),(3), and (5) are true.

(B) Only (1),(3), and (4) are true.

(C) Only (2),(3), and (4) are true.

(D) Only (2),(3), and (5) are true.

(E) Only (1),(2), and (5) are true.

I know that (1) is true, (2) is false and (3) is true. What is the difference between (4) and (5) though? How do I know which one is correct? Apparently statement (4) is the correct one but why is that?

If you are dealing with linear algebra, the chances are high that you will encounter various questions related to matrix transformation. Turning to matrix transformation examples, you will also encounter various geometric transformations, yet these will always be based on algebraic analysis and calculations. The answers that we have presented to various challenges will help you to compare our solutions with your unique matrix transformation example that deals with linear transformation and mapping. Visual assistance is also included and will be essential to see how these are built with the help of the column vectors.