Recent questions in Matrix transformations
Cory Patrick 2022-06-22

Can someone please explain the difference between a matrix and linear transformation. I know that they can be the same thing, but have been told that they aren't the same thing necessarily. So my question is what is the difference?

Reed Eaton 2022-06-22

What does a symmetric matrix transformation do, geometrically?Need some visual intuition behind what exactly a symmetric matrix transformation does. In a $2×2$ and $3×3$ vector space, what are they generally?

Davon Irwin 2022-06-21

We have a linear transformation $T:{M}_{2×2}\left(F\right)\to F$ by $T\left(A\right)=tr\left(A\right)$.We want to compute the matrix representation $\left[T\right]$ from $\alpha$ to $\gamma$ coordinates.${M}_{2×2}$ has the standard ordered basis "$\alpha$" for a $2×2$ matrix. $F$ has the standard basis "$\gamma$" for a scalar.My understanding is that any matrix $A$ in the space ${M}_{2×2}$ can be represented by the linear combination:$a{\alpha }_{1}+b{\alpha }_{2}+c{\alpha }_{3}+d{\alpha }_{4}$and the $tr\left(A\right)$ can be written as:$\left(a+c\right)\gamma .$I'm not sure how to get the matrix representation from this.

Leonel Contreras 2022-06-21

Find the standard matrix for the following composition in ${\mathbb{R}}^{\mathbb{2}}$:A reflection about the $x$-axis followed by a rotation of $\frac{\pi }{6}$

boloman0z 2022-06-21

Let $\psi$ be a linear operator on $V$, a vector space of dimension two. How do I show, given $\psi$ is not a scalar multiple of the identity, that there is a $v$ such that $v,\psi \left(v\right)$ forms a basis, and further how do I write the transformation as a matrix with respect to that matrix? For the matrix, the first column would be 0,1, but I don't know what the second column would be.

Roland Waters 2022-06-20

Let $B=\left\{{v}_{1},...,{v}_{n}\right\}$ and $C=\left\{{w}_{1},...,{w}_{n}\right\}$ be bases to V. Suppose: ${w}_{i}={m}_{i1}{v}_{1}+...+{m}_{in}{v}_{n}$ for ${m}_{ij}\in F,1\le i,j\le n$ $M$ is an invertible matrix whose $i,j$ member is ${m}_{ij}$need to express the transformation matrix, N, from B to C by M. ($\mathrm{\forall }v\in V\left[v{\right]}_{C}=N\left[v{\right]}_{B}$)Is this true?$\left({w}_{1}\dots {w}_{n}{\right)}^{-1}\left({v}_{1}\dots {v}_{n}\right)=N\phantom{\rule{0ex}{0ex}}\left(\left({v}_{1}\dots {v}_{n}\right){M}^{t}{\right)}^{-1}\left({v}_{1}\dots {v}_{n}\right)=N\phantom{\rule{0ex}{0ex}}\left({M}^{t}{\right)}^{-1}\left({v}_{1}\dots {v}_{n}{\right)}^{-1}\left({v}_{1}\dots {v}_{n}\right)=N\phantom{\rule{0ex}{0ex}}\left({M}^{-1}{\right)}^{t}=N$

Dale Tate 2022-06-20

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

hawatajwizp 2022-06-20

Let $\phi :K\left[x{\right]}_{\le n}\to K\left[x{\right]}_{\le n-1}$ with $\phi$ the linear transformation defind by $\phi \left(f\right)={f}^{\prime }$. Select a base and find the matrix of the linear transformation.I took the standard basis for grade-n polynomials:$B=<1,x,{x}^{2},\dots ,{x}^{n}>,\phi \left(B\right)=\phi \left(1,x,{x}^{2},...,{x}^{n}\right)=\left(0,2x,...,n{x}^{n-1}\right)$So, the matrix is $\left[\begin{array}{c}0\\ 2x\\ \dots \\ n{x}^{n-1}\end{array}\right]$?

watch5826c 2022-06-20

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

vittorecostao1 2022-06-19

Showing that the matrix transformation $T\left(f\right)=x\ast {f}^{\prime }\left(x\right)+{f}^{″}\left(x\right)$ is linear

Feinsn 2022-06-17

Consider a linear transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ satisfyingThe 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 $T$?

Quintin Stafford 2022-06-17

Determine the Linear Transformation MatrixThe 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 $S\left(x\right)=R\left(x\right)+3x.$how to get the Matrix $S$ from the definition above.

glycleWogry 2022-06-16

How to solve matrix transformation of a vector?Given two non-zero vectors $a=\left({a}_{1},{a}_{2}\right)$ and $b=\left({b}_{1},{b}_{2}\right)$ we define $\sim \left(a,b\right)$ by $\sim \left(a,b\right)={a}_{1}{b}_{2}-{a}_{2}{b}_{1}$. 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 $2×2$ matrix $M$ such that $P$ and $Q$ are the images of $A$ and $B$ under transformation represented by $M$.

George Bray 2022-06-16

I know that the matrix representation of the linear transformation:${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?

April Bush 2022-06-16

For each $m\in \left\{1,2,...,n\right\}$ , is there a transformation ${\varphi }_{m}$ that I can apply to a matrix $M\in {\mathbb{R}}^{n×n}$ such that the ${m}^{th}$ largest eigenvalue ${\lambda }_{m}$ of $M$ is the smallest eigenvalue ${\lambda }_{n}^{\prime }$ of the matrix ${M}^{\prime }={\varphi }_{m}\left(M\right)$?

enrotlavaec 2022-06-16

Here's the question: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?

Misael Matthews 2022-06-16

Suppose that $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is a linear transformation. Prove that there exists an $m×n$ matrix $A$ such that for all $x\in {\mathbb{R}}^{n}$, $T\left(x\right)=Ax$. In other words, $T$ is the "matrix transformation" associated with $A$.

Dwllane4 2022-06-16

In Sakurai's Modern quantum mechanics it is said that the rotation matrix in three dimensions that changes one set of unit basis vectors $\left(x,y,z\right)$ into another set $\left({x}^{\prime },{y}^{\prime },{z}^{\prime }\right)$ can be written as$\left[\begin{array}{ccc}x{x}^{\prime }& x{y}^{\prime }& x{z}^{\prime }\\ y{x}^{\prime }& y{y}^{\prime }& y{z}^{\prime }\\ z{x}^{\prime }& z{y}^{\prime }& z{z}^{\prime }\end{array}\right]$But shouldn't it be the transpose of matrix given above as the transformation matrix is given by coordinates of transformations of bases ?

excluderho 2022-06-15

Let $X$ be a $m×n$ (m: number of records, and n: number of attributes) normalized dataset (between 0 and 1). Denote $Y=XR$, where $R$ is an $n×p$ matrix, and $p. I understand if $R$ was drawn randomly from Gaussian distribution, e.g.,$N\left(0,1\right)$ then the transformation preserve the Euclidean distances between instances (all of the pairwise distances between the points in the feature space will be preserved). But what if $R\sim U\left(0,1\right)$, does the transformation still preserve the distance between instances?

Santino Bautista 2022-06-15

Which of the following statements are true?(1) For $\theta ϵ\mathbb{R}$ fixed, $R:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{2}},R\left(x,y\right)=\left(x\mathrm{cos}\left(\theta \right)-y\mathrm{sin}\left(\theta \right),x\mathrm{sin}\left(\theta \right)+y\mathrm{cos}\left(\theta \right)\right)$ is linear.(2) The transformation $T:{\mathbb{R}}^{\mathbb{2}}\to {\mathbb{R}}^{\mathbb{2}},T\left(x,y\right)=\left(1+{x}^{2},1+{y}^{2}\right)$ is linear.(3) For $n\ge 1$, every linear transformation $L:R:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ satisties $L\left(\mathbf{0}\right)=\mathbf{0}$(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.