# Matrix Transformation Examples and Practice Problems

Recent questions in Matrix transformations

## Lets say, there is a transformation: $T:{\mathrm{\Re }}^{n}\to {\mathrm{\Re }}^{m}$ transforming a vector in $V$ to $W$. Now the transformation matrix,$A=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& ...& {a}_{1n}\\ {a}_{21}& {a}_{22}& ...& {a}_{2n}\\ .& .& .\\ .& .& .\\ .& .& .\\ {a}_{n1}& {a}_{n2}& ...& {a}_{n3}\end{array}\right]$The basis vectors of $V$ are ${v}_{1},{v}_{2},{v}_{3}...,{v}_{n}$ which are all non standard vectors and similarly ${w}_{1},{w}_{2},...,{w}_{m}$My question is, in the absence of the basis vectors being standard vectors what is the procedure of finding $T$

aangenaamyj 2022-07-07

## If $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is a matrix transformation, does $Tn,m=n, orn>m?$Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?

tripes3h 2022-07-07

## Suppose $T:V\to V$ is the identity transformation.If B is a basis of V, then the matrix representation of $\left[T{\right]}_{B}^{B}=\left[{I}_{n}\right]$.Let's say C is also a basis of V, then it is clear that $\left[T{\right]}_{C}^{B}\ne \left[{I}_{n}\right]$However, I was taught that matrices representing the same linear transformation in different bases are similar, and the only matrix similar to ${I}_{n}$ is ${I}_{n}$. Thus, $\left[T{\right]}_{C}^{B}$ and $\left[T{\right]}_{B}^{B}$ are not similar.Can anyone clear what seems to be a contradiction?

aggierabz2006zw 2022-07-07

## Given a matrix (3 * 1)$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$how can I obtain a matrix 3 * 3 of$\left[\begin{array}{ccc}a-a& a-b& a-c\\ b-a& b-b& b-c\\ c-a& c-b& c-c\end{array}\right]$

Blericker74 2022-07-06

## Matrix, that projects, any point of the xy-plane, on the line$y=4x$The solution should be:$T=\left(\begin{array}{cc}0.06& 0.235\\ 0.235& 0.94\end{array}\right)$But somehow i dont know how to get this solution?

pouzdrotf 2022-07-06

## Is the result of a matrix transformation equivalent to the that of that same matrix but orthonormalized?Say we have the transformation matrix $A$ of full rank such that ${A}^{-1}\ne {A}^{t},$, i.e., the matrix $A$ consists of linearly independent vectors which aren't orthogonal to each other, a vector v, and the orthonormalized transformation matrix ${A}^{\prime }.$ Is it true that $Av={A}^{\prime }v?$ And if not, is this unimportant?

Jonathan Miles 2022-07-06

## Question that asks me to describe a vector x that satisfies:$T\left(x\right)=\left[\begin{array}{c}-8\\ 9\\ 2\end{array}\right]$Gven matrix:$A=\left[\begin{array}{ccc}1& 3& 1\\ -2& 1& 5\\ 0& 2& 2\end{array}\right]$also aware that T(x) = Ax. I would like to know the general process for finding what x is when given the output vector and a matrix to be multiplied by the unknown input vector x.$\left[\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$I have tried putting the augmented matrix in reduced row echelon form above, but I am not sure where to go from there.

Aganippe76 2022-07-06

## Take, for instance, the following matrix:$\left[\begin{array}{cc}12& 5\\ 5& -12\end{array}\right]$How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation?

gorgeousgen9487 2022-07-06

## $A=\left(\begin{array}{cc}k& -2\\ 1-k& k\end{array}\right)\text{, where k is a constant}$

Logan Wyatt 2022-07-05

## The Transformation of A is defined on the space ${\mathcal{P}}_{2}$ of polynomials $p$ such that $\mathrm{deg}\left(p\right)\le 2$ by $Ap\left(t\right)={p}^{\prime }\left(t\right)$. Find the matrix of this transformation in the basis $\left\{1,t,{t}^{2}\right\}$. What is $Ker\left(A\right)$?

vasorasy8 2022-07-05

## $T\left({e}_{1}\right)=T\left(1,0\right)=\left(\mathrm{cos}\theta ,\mathrm{sin}\theta \right)$and$T\left({e}_{2}\right)=T\left(0,1\right)=\left(-\mathrm{sin}\theta ,\mathrm{cos}\theta \right)$and$A=\left[T\left({e}_{1}\right)|T\left({e}_{2}\right)\right]=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$When I rotate a vector $\left[\begin{array}{c}x\\ y\end{array}\right]$ I get$\left[\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right]=\left[\begin{array}{c}x\cdot \mathrm{cos}\theta \phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}y\cdot \mathrm{sin}\theta \\ x\cdot \mathrm{sin}\theta \phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}y\cdot \mathrm{cos}\theta \end{array}\right]$Correct me if I'm wrong, but I thought that column 1 of A $\left[\begin{array}{c}\mathrm{cos}\theta \\ \mathrm{sin}\theta \end{array}\right]$, holds the 'x' values and column 2 holds the 'y' values. What I'm confused about is why does x' contain both an x component and a y component?

malalawak44 2022-07-05

## Does a non-invertible matrix transformation "really" not have an inverse?

Shea Stuart 2022-07-05

## Find matrix representation of transformationGiven two lines ${l}_{1}:y=x-3$ and ${l}_{2}:x=1$ find matrix representation of transformation $f$(in standard base) which switch lines each others and find all invariant lines of $f$.

Frank Day 2022-07-04

## Suppose $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is a matrix transformation. I want to show that$T\left(\stackrel{\to }{0}\right)=\stackrel{\to }{0}$Since $T$ is a matrix transformation, then for every $\stackrel{\to }{x}\in {\mathbb{R}}^{n}$ there exists a unique $m×n$ matrix $A$ such that$T\left(\stackrel{\to }{x}\right)=A\cdot \stackrel{\to }{x}$If we let $\stackrel{\to }{x}=\stackrel{\to }{0}$, where this zero vector has dimensions $n×1$, then$T\left(\stackrel{\to }{0}\right)=A\cdot \stackrel{\to }{0}=\stackrel{\to }{0}$where the $\stackrel{\to }{0}$ on the right hand side of the equation is an $m×1$ vector.Hence, every transformation maps the zero vector in ${\mathbb{R}}^{n}$ to the zero vector in ${\mathbb{R}}^{m}$Are there any problems with the proof?

uplakanimkk 2022-07-03

## Using matrix methods, how to find the image of the point (1,-2) for the transformations? 1) a dilation of factor 3 from the x-axis 2) reflection in the x-axis

prirodnogbk 2022-07-03

## Consider the $m×m$-matrix $B$, which is symmetric and positive definite (full rank). Now this matrix is transformed using another matrix, say $A$, in the following manner: $AB{A}^{T}$. The matrix A is $n×m$ with $n. Furthermore the constraint $rank\left(A\right) is imposed.My intuition tells me that $AB{A}^{T}$ must be symmetric and positive semi-definite, but what is the mathematical proof for this? (why exactly does the transformation preserve symmetry and why is it that possibly negative eigenvalues in $A$ still result in the transformation to be PSD? Or is my intuition wrong)?

mistergoneo7 2022-07-02

## By conjugate linear transformation, I mean under scalar multiplication instead of $C\left(af\right)=aC\left(f\right)$, I would have $C\left(af\right)=\overline{a}C\left(f\right)$, where a is $a$ constant complex number, and $C$ is the transformation.

Nylah Hendrix 2022-07-02