Recent questions in Matrix transformations

Matrix transformations
Answered

Izabella Ponce
2022-06-26

$A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$

is the matrix representation of a linear transformation

$T:{P}_{2}(x)\to {P}_{2}(x)$

with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+{x}^{2}\}$ then find T. What is the procedure to solve it?

Matrix transformations
Answered

Quintin Stafford
2022-06-26

rotation by 30 degrees about x-axis

translation by 1, -1, 4 in x, y and z, respectively

rotation by 45 degrees about y axis

Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?

Matrix transformations
Answered

Llubanipo
2022-06-25

Matrix transformations
Answered

George Bray
2022-06-25

2) Suppose $V$ is the direct sum of ${W}_{1}$ and ${W}_{2}$, with ${W}_{1},{W}_{2}$ both T-invariant. If $B=({u}_{1},{u}_{2},\dots ,{u}_{n},{z}_{1},{z}_{2},\dots ,{z}_{n})$ is a basis for $V$ with the $u$'s being a basis for ${W}_{1}$ and the $z$'s being a basis for ${W}_{2}$, what does the matrix of $T$ with respect to $B$ look like?

Matrix transformations
Answered

Semaj Christian
2022-06-25

Matrix transformations
Answered

taghdh9
2022-06-24

two bases:

$A=\{{v}_{1},{v}_{2},{v}_{3}\}$ and $B=\{2{v}_{1},{v}_{2}+{v}_{3},-{v}_{1}+2{v}_{2}-{v}_{3}\}$

There is also a linear transformation: $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$

Matrix in base $A$:

${M}_{T}^{A}=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 1& 1& 0\end{array}\right]$

Now I am to find matrix of the linear transformation $T$ in base $B$.

I have found two transition matrixes (from base $A$ to $B$ and from $B$ to $A$):

${P}_{A}^{B}=\left[\begin{array}{ccc}2& 0& -1\\ 0& 1& 2\\ 0& 1& -1\end{array}\right]$

$({P}_{A}^{B}{)}^{-1}={P}_{B}^{A}=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{6}& \frac{-1}{6}\\ 0& \frac{1}{3}& \frac{2}{3}\\ 0& \frac{1}{3}& \frac{-1}{3}\end{array}\right]$

How can I find ${M}_{T}^{B}$?

Matrix transformations
Answered

Ezekiel Yoder
2022-06-24

$\begin{array}{rcl}M& =& {I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}\\ & =& {I}_{n}-AB{\left({B}^{T}{A}^{T}AB\right)}^{-1}{B}^{T}{A}^{T},\end{array}$

since $AB$ is basically the linear transformation of matrix $A.$

Matrix transformations
Answered

protestommb
2022-06-24

$\begin{array}{rcl}M& =& {I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}\\ & =& {I}_{n}-AB{\left({B}^{T}{A}^{T}AB\right)}^{-1}{B}^{T}{A}^{T},\end{array}$

since $AB$ is basically the linear transformation of matrix $A.$

Matrix transformations
Answered

hawatajwizp
2022-06-24

Matrix transformations
Answered

Mohamed Mooney
2022-06-24

$F(X)={X}^{T}$

on $V={M}_{3,3}(\mathbb{R})$, apparently the matrix of $F$ is a $9\times 9$ matrix. How can this be possible? Isn't the definition that

$F(X)=AX?$

, so if $A$ is the $9\times 9$ matrix, we can't multiply a $9\times 9$ matrix with a $3\times 3$, can we?

Matrix transformations
Answered

Petrovcic2x
2022-06-24

Here's my thinking so far:

Since the standard matrix for reflections in xy is

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Similarly, standard matrix for orthogonal projection in the xz plane is

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$

I could multiply

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\ast \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

to yield

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Could someone confirm for me if this is a valid approach?

Matrix transformations
Answered

Abram Boyd
2022-06-24

$\left[\begin{array}{ccc}-1& -2& -5\\ 0& 2& 2\\ 2& 2& 4\end{array}\right]$

a) Determine the transformation matrix $A$ with respect to the basis $\{x+y,y+z,z+x\}.$

b)Determine if A is a linear transformation.

$A$

What is the process in determining the transformation matrix with respect to a new basis when you already have a transformation matrix with respect to another?

Matrix transformations
Answered

Celia Lucas
2022-06-23

Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the positive-definite aspect in the resuling matrix ?

Matrix transformations
Answered

veirarer
2022-06-23

$T({x}_{1},{x}_{2})=(3{x}_{1},{x}_{1}+2{x}_{2})$

Is this a matrix transformation such that $T(\overrightarrow{x})=\left[\begin{array}{c}3{x}_{1}\\ {x}_{1}+2{x}_{2}\end{array}\right]$?

If so, then am I correct to say the matrix must be $A=\left[\begin{array}{cc}3& 0\\ 1& 2\end{array}\right]$ and $\overrightarrow{x}=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$, so

$T(\overrightarrow{x})=\left[\begin{array}{cc}3& 0\\ 1& 2\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right];\text{}T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$

Matrix transformations
Answered

protestommb
2022-06-23

I tried to solve the linear matrix equation AX = B to get the matrix transformation A as A = B X^T (XX^T)^{-1} , where X is the matrix to be transformed as such, and B is the desired output matrix (X with only its first nonzero element in each row) but I look for a more straightforward and possibly more elegant way to do it. I appreciate creative answers!

Matrix transformations
Answered

Makayla Boyd
2022-06-23

Matrix transformations
Answered

gvaldytist
2022-06-23

Matrix transformations
Answered

Jeramiah Campos
2022-06-23

$T(x,y,z)=(\u20134x+3y+z,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x\u20135y,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6y+7z).$

Find the standard matrix for $T$, and use that matrix to find $T(1,\u20133).$.

Matrix transformations
Answered

Emmy Knox
2022-06-22

Matrix transformations
Answered

George Bray
2022-06-22

A) T(x,y,z)=(0,0)

B) T(x,y,z)=(1,-1)

C) T(x,y)=(xy,y)

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.