# Matrix transformation questions and answers

Recent questions in Matrix transformations
Izabella Ponce 2022-06-26

### If$A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$is the matrix representation of a linear transformation$T:{P}_{2}\left(x\right)\to {P}_{2}\left(x\right)$with respect to the bases $\left\{1-x,x\left(1-x\right),x\left(1+x\right)\right\}$ and $\left\{1,1+x,1+{x}^{2}\right\}$ then find T. What is the procedure to solve it?

Quintin Stafford 2022-06-26

### If I am required to compute the full transformation matrix compromising of the following sequence of operations:rotation by 30 degrees about x-axistranslation by 1, -1, 4 in x, y and z, respectivelyrotation by 45 degrees about y axisCan I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?

Llubanipo 2022-06-25

### I know that matrix multiplication is not commutative. And yet, I am taught that transformation matrices can be composed to arrive at a single transformation. So then, suppose that I have 3 matrices - each representing a rotation about each axis of a 3D coordinate system. How can I be sure that I arrive at the correct result, if the order matters?

George Bray 2022-06-25

### 1) Suppose $B=\left({w}_{1},{w}_{2},\dots ,{w}_{k},{v}_{1},\dots ,{v}_{n-k}\right)$ is a basis for $V$ and $W$ = Span $\left({w}_{1},\dots ,{w}_{k}\right)$ is T-invariant. What does the matrix of $T$ with respect to $B$ look like?2) Suppose $V$ is the direct sum of ${W}_{1}$ and ${W}_{2}$, with ${W}_{1},{W}_{2}$ both T-invariant. If $B=\left({u}_{1},{u}_{2},\dots ,{u}_{n},{z}_{1},{z}_{2},\dots ,{z}_{n}\right)$ 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?

Semaj Christian 2022-06-25

### to with the conditions that $F\left({p}_{0}\right)=f$, $F\left({p}_{1}\right)=g$, $F\left({p}_{2}\right)=h$ where $f\left(x\right)={x}^{2}+3$, $g\left(x\right)={x}^{2}-x$, $h\left(x\right)=2+x$. Find the transformation matrix for $F$ in the basis $\left({p}_{0},{p}_{1},{p}_{2}\right)$

taghdh9 2022-06-24

### Linear transformation and its matrixtwo bases:$A=\left\{{v}_{1},{v}_{2},{v}_{3}\right\}$ and $B=\left\{2{v}_{1},{v}_{2}+{v}_{3},-{v}_{1}+2{v}_{2}-{v}_{3}\right\}$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]$$\left({P}_{A}^{B}{\right)}^{-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 ${M}_{T}^{B}$?

Ezekiel Yoder 2022-06-24

### struggling on the properties of the idempotent matrix, namely for any $n×n$ matrix, ${A}^{2}=A.$. The projection matrix defined by $M={I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}$ is an idempotent matrix. The question is, for any given $n×m$ ($n>m$) matrix $B,$, do we have$\begin{array}{rcl}M& =& {I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}\\ & =& {I}_{n}-AB{\left({B}^{T}{A}^{T}AB\right)}^{-1}{B}^{T}{A}^{T},\end{array}$since $AB$ is basically the linear transformation of matrix $A.$

protestommb 2022-06-24

### struggling on the properties of the idempotent matrix, namely for any $n×n$ matrix, ${A}^{2}=A.$. The projection matrix defined by $M={I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}$ is an idempotent matrix. The question is, for any given $n×m$ ($n>m$) matrix $B,$, do we have$\begin{array}{rcl}M& =& {I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}\\ & =& {I}_{n}-AB{\left({B}^{T}{A}^{T}AB\right)}^{-1}{B}^{T}{A}^{T},\end{array}$since $AB$ is basically the linear transformation of matrix $A.$

hawatajwizp 2022-06-24

### A linear transformation acts on ${R}^{2}$ by first doubling the x-coordinate and then rotating the plane by an angle of $\pi /2$ (${90}^{\circ }$) counter-clockwise. What is the corresponding matrix to this linear transformation?

Mohamed Mooney 2022-06-24

### For a transformation$F\left(X\right)={X}^{T}$on $V={M}_{3,3}\left(\mathbb{R}\right)$, apparently the matrix of $F$ is a $9×9$ matrix. How can this be possible? Isn't the definition that$F\left(X\right)=AX?$, so if $A$ is the $9×9$ matrix, we can't multiply a $9×9$ matrix with a $3×3$, can we?

Petrovcic2x 2022-06-24

### Let ${T}_{1}$ be a reflection of ${\mathbb{R}}^{3}$ in the xy plane, ${T}_{2}$ is a reflection of ${\mathbb{R}}^{3}$ in the xz plane. What is the standard matrix of transformation ${T}_{2}{T}_{1}$?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?

Abram Boyd 2022-06-24

### Let $\left\{x,y,z\right\}$ be the basis of ${\mathbb{R}}^{3}$ and $A:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ is a linear operator given by the matrix:$\left[\begin{array}{ccc}-1& -2& -5\\ 0& 2& 2\\ 2& 2& 4\end{array}\right]$a) Determine the transformation matrix $A$ with respect to the basis $\left\{x+y,y+z,z+x\right\}.$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?

Celia Lucas 2022-06-23

### Matrix transformation conserving the "positive semi-definite" aspectLet'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 ?

veirarer 2022-06-23

### How do interpret the notation$T\left({x}_{1},{x}_{2}\right)=\left(3{x}_{1},{x}_{1}+2{x}_{2}\right)$Is this a matrix transformation such that $T\left(\stackrel{\to }{x}\right)=\left[\begin{array}{c}3{x}_{1}\\ {x}_{1}+2{x}_{2}\end{array}\right]$?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 $\stackrel{\to }{x}=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$, so

protestommb 2022-06-23

### Need to find a matrix transformation that takes a (non-square) matrix and within each of its rows keeps the first nonzero element in that row and zeros out the rest of the entries within that row.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!

Makayla Boyd 2022-06-23

### Let $A$ be an $m×n$ matrix, and $\mathbf{x}$ be a vector in ${\mathbb{R}}^{n}$. A transformation $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is linear if and only if it is a matrix transformation of the form ${T}_{A}\left(\mathbf{x}\right)=A\mathbf{x}$, where $T={T}_{A}$

gvaldytist 2022-06-23

### Give an example of non linear transformation matrix? What is the difference between linear and non linear transformation matrix?

Jeramiah Campos 2022-06-23

### Consider the matrix transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ defined by$T\left(x,y,z\right)=\left(–4x+3y+z,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x–5y,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6y+7z\right).$Find the standard matrix for $T$, and use that matrix to find $T\left(1,–3\right).$.

Emmy Knox 2022-06-22

### Consider the transformation $T:{P}_{2}\to {P}_{2}$, where ${P}_{2}$ is the space of second-degree polynomials matrices, given by $T\left(f\right)=f\left(-1\right)+f\prime \left(-1\right)\left(t+1\right)$. Find the matrix for this transformation relative to the standard basis $\mathfrak{A}=\left\{1,t,{t}^{2}\right\}$. Can someone explain to me how to find the matrix of the transformation

George Bray 2022-06-22

### Determine whether T is a matrix transformationA) 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.