# Matrix Transformation Examples and Practice Problems

Recent questions in Matrix transformations
Ciara Mcdaniel 2022-07-01

## Finding inverse matrix ${A}^{-1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A.$A=\left(\begin{array}{cccc}1& 3& 9& 27\\ 3& 3& 9& 27\\ 9& 9& 9& 27\\ 27& 27& 27& 27\end{array}\right)$$B=\left(\begin{array}{cccc}1& 3& 9& 27\\ 1& 1& 3& 9\\ 1& 1& 1& 3\\ 1& 1& 1& 1\end{array}\right)$I found that that ${B}^{-1}$ is${B}^{-1}=\left(\begin{array}{cccc}-\frac{1}{2}& \frac{3}{2}& 0& 0\\ \frac{1}{2}& -2& \frac{3}{2}& 0\\ 0& \frac{1}{2}& -2& \frac{3}{2}\\ 0& 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right)$I don't know how to continue. What rule do I use to find ${A}^{-1}$?

Cristopher Knox 2022-07-01

## Prove that pre-multiplying a matrix ${A}_{m}$ by the elementary matrix obtained with any matrix elementary line transformation ${I}_{m}\underset{{l}_{1}↔{l}_{2}}{⟶}E$ is the same as applying said elementary line transformation on the matrix ${A}_{m}$

Kristen Stokes 2022-07-01

## $T\left(p\left(x\right)\right)={\int }_{0}^{1}p\left(x\right)dx.$(a) Show $T$ is a linear transformation.(b) Compute $\mathcal{N}\left(T\right).$ Is $T$ one-to-one?(c) Show that $T$ is onto.(d) Let $B$ be the standard basis for ${\mathcal{P}}_{2}$ and let ${B}^{\mathrm{\prime }}=\left\{1\right\}$ be a basis for $\mathbb{R}$. Find $\left[T{\right]}_{B}^{{B}^{\mathrm{\prime }}}$.(e) Use the matrix found in part (d) to compute $T\left(-{x}^{2}-3x+2\right)$

spockmonkey40 2022-06-30

## Find the matrix A for the linear transformation T relative to the bases $B=\left\{1,x,{x}^{2}\right\}$ and $B\prime =\left\{1,x,{x}^{2},{x}^{3}\right\}$ such that $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$

kokoszzm 2022-06-30

## Find the transformation matrix:$F\left(v\right)=\frac{{d}^{2}v}{d{v}^{2}}$Basis: $1,x,{x}^{2},{x}^{3}$ and ${\mathbb{R}}_{3}\left[x\right]$ - the set of all third degree polynomials of variable $x$ over $\mathbb{R}$ Assume that all coefficients of the polynomials are $1$The first thing that springs to my mind is to calculate this derivative by hand, and so we got$\frac{{d}^{2}y}{d{y}^{2}}=2+6x$Now, we need to put these values - $2$ and $6$ in such a matrix that - when multiplied by the basis vector -will give us $2+6x$ But there are many ways I can think of, for example$\left[\begin{array}{cccc}0& 0& 2& 0\\ 0& 0& 0& 6\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$Or maybe$\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 6& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$Because both of them, when multiplied by$\left[\begin{array}{c}1\\ x\\ {x}^{2}\\ {x}^{3}\end{array}\right]$Will give the correct answer. Thus, what is the correct way to solve this?

Jameson Lucero 2022-06-29

## Consider in ${\mathbb{R}}^{2}$ the set of points satisfying the equation (you can use matlab for this). How are these points transformed by the following matrix:$\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]$[Show the transformed set and plot it – you can use matlab again]I could be mistaken but would the matrix be$\left[\begin{array}{cc}2& -1-2\end{array}\right]?$After that I am very stuck. Thanks

veneciasp 2022-06-29

## If the transformation is from ${\mathbb{R}}^{3}\to \mathbb{R}$ is $T\left\{a,b,c\right\}={\int }_{0}^{\pi }2a{e}^{t}+2b\mathrm{sin}\left(t\right)+3c\mathrm{cos}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$How to find the standard matrix?

bandikizaui 2022-06-29

## In an affine transformation $x↦Ax+b$, $b$ represents the translation; but what does the matrix A represent exactly? In a 2D example, $A$ is a $2×2$ matrix, but what does each term represent?

Devin Anderson 2022-06-29

## If I take a particular number, for example 21, and write down all possible matrices whose determinant is 21, will all these matrices represent the same transformation with different system of bases? Consider all endomorphisms.

Makayla Boyd 2022-06-28

## Transformation matrix for matrix indices to cartesian coordinates(1,1) (1,2) (1,3)(2,1) (2,2) (2,3)(3,1) (3,2) (3,3)This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation would be:(-1,1) (0,1) (1,1)(-1,0) (0,0) (1,0)(-1,1) (0,-1) (1,-1)Say, I have any square matrix with dimension-N, where N is odd. I need a generic transformation matrix such that I can get a vector as cartesian coordinates from matrix indices. Does such a function already exist? How should I go ahead in implementing this?

Dania Mueller 2022-06-27

## Let $v\in {T}_{2}\left(V\right)$ be a bilinear form over finite space V. Let T be a Linear transformation $V\to V$. We define: ${v}_{T}\left(x,y\right)=v\left(T\left(x\right),y\right)$Assuming $v$ is nondegenerate, let us have another bilinear form $\xi \in {T}_{2}\left(V\right)$. Prove that there exists exactly one transformation $T$ so $\xi ={v}_{T}$.

Poftethef9t 2022-06-27

## Let's say we define the orientation of the ellipsoid from its major axis (the largest axis of the ellipsoid). Assuming the 3 axes of the ellipsoid to be on the three coordinates with lengths of $a$, $b$ and $c$ along each axis (with $a\ge b$ and $a\ge c$), then only a single affine transformation (to a sphere) is necessary:$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]$Now, if we first rotate the major axis by $\theta$ from the first axis towards the second axis, and then rotate it by $\varphi$ from the (rotated) first axis towards the third axis, the combined affine transformation becomes:$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\varphi & 0& -sin\varphi & 0\\ 0& 1& 0& 0\\ sin\varphi & 0& cos\varphi & 0\\ 0& 0& 0& 1\end{array}\right]$Is the multiplied matrix (from left to right) the correct affine transformation that must go into the equations in the links above?

Feinsn 2022-06-26

## Suppose $T$ is an element of $L\left({P}_{3}\left(\mathbb{R}\right),{P}_{2}\left(\mathbb{R}\right)\right)$ is the differentiation map defined by $Tp={p}^{\prime }$. Find a basis of ${P}_{3}\left(\mathbb{R}\right)$ and a basis of ${P}_{2}\left(\mathbb{R}\right)$ such that the matrix of T with respect to these basis is $\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

taghdh9 2022-06-26

## I have a rectangle with center $\left({x}_{1},{y}_{1}\right)$ and sides $a,b$ where side $a$ is parallel to axis $Ox$. I want to find a transformation matrix that:a) converts this rectangle into a square with the same center and side $d$.b) reflects the rectangle with mirror axis the line $y=sx+c$.

Boilanubjaini8f 2022-06-26

## A linear transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{2}$ whose matrix is $\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -3.5+k\end{array}\right)$is onto if and only if $k\ne$

George Bray 2022-06-26

## Let $V$ be inner product space.Let ${e}_{1},...,{e}_{n}$ an orthonormal basis for $V$Let ${z}_{1},...,{z}_{n}$ an orthonormal basis for $V$I have to show that the matrix represents the transformation matrix between ${e}_{1},...,{e}_{n}$ to ${z}_{1},...,{z}_{n}$ is unitary.

Izabella Ponce 2022-06-26