Matrix Transformation Examples and Practice Problems

Recent questions in Matrix transformations
Chelsea Lamb 2022-09-26

If the matrix of a linear transformation $:{\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to geometrically interpret the transformation in a nice/simple way?

Julia Chang 2022-09-25

Given a matrix $Y\in {\mathbb{R}}^{m×n}$. Find a transformation matrix $\mathrm{\Theta }\in {\mathbb{R}}^{n×p}$ such that$\frac{1}{m}{\mathrm{\Theta }}^{T}{Y}^{T}Y\mathrm{\Theta }={I}_{p×p},$where 𝐼𝑝×𝑝 is identity matrix.My attempt: $\frac{1}{\sqrt{m}}Y\mathrm{\Theta }$ is orthogonal matrix and tried to find $\mathrm{\Theta }$ satisfies it but that doesn't work.

Celinamg8 2022-09-20

Vector, $u:=\left[{u}_{1},\dots ,{u}_{n}{\right]}^{\mathrm{T}}$. I am trying to find a coordinate transformation matrix, $Q\in {\mathbb{R}}^{n×n}$, which is nonsingular, satisfying:$\begin{array}{r}\left[\begin{array}{c}0\\ ⋮\\ 0\\ ||u||\end{array}\right]=Qu.\end{array}$

ahmed zubair2022-09-14

A trust fund has $200,000 to invest. Three alternative investments have been identified, earning income of 10 percent, 7 percent and 8 percent respectively. A goal has been set to earn an annual income of$16,000 on the total investment. One condition set by the trust is that the combine investment in alternatives 2 and 3 should be triple the amount invested in alternative 1. Determine the amount of money, which should be invested in each option to satisfy the requirements of the trust fund. Solve by Gauss- Jordon method What are the equations formed in this question?

nar6jetaime86 2022-09-13

Findthe Matrix T of the following linear transformation$T\phantom{\rule{mediummathspace}{0ex}}:\phantom{\rule{mediummathspace}{0ex}}R2\left(x\right)\phantom{\rule{mediummathspace}{0ex}}->R2\left(x\right)\phantom{\rule{mediummathspace}{0ex}}defined\phantom{\rule{mediummathspace}{0ex}}by\phantom{\rule{mediummathspace}{0ex}}T\left(a{x}^{2}+bx+c\right)=2ax+b$

tamola7f 2022-09-13

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal matrix represents a rotation (and possibly a reflection). Is it something similar about triangular matrices? Do they represent any specific type of transformation?

Julius Blankenship 2022-09-12

Consider ${\mathbb{K}}^{n}$, ${\mathbb{K}}^{m}$, both with the $||.|{|}_{1}$-norm, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.Let be the operator norm of a linear transformation $T:{\mathbb{K}}^{n}\to {\mathbb{K}}^{m}$.Show that the operator norm of the linear transformation $T$ is also given by:$||T||=max\left\{\sum _{i=1}^{m}|{a}_{ij}|,1\le j\le n\right\}=:||A|{|}_{1}$where $A$ is the transformation matrix of $T$ and ${a}_{ij}$ it's entry in the $i$-th row and $j$-the column.

tuzkutimonq4 2022-09-11

Convert the following matrix:$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$To the following:$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 0& 1& 0& 1\end{array}\right]$

engausidarb 2022-09-11

Real symmetric matrices ${S}_{ij}$ can always be put in a standard diagonal form ${s}_{i}{\delta }_{ij}$ under an orthogonal transformation. Similarly, real antisymmetric matrices ${A}_{ij}$ can always be put in a standard band diagonal form with diagonal matrix entries ${a}_{i}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ (with a $0$ diagonal entry when the dimension of the matrix is odd), again under an orthogonal transformation.

moidu13x8 2022-09-09

How to find a matrix of linear transformation $f:{R}^{n}\to Ma{t}^{\left(n,n\right)}$.Let's say we do $f:{R}^{2}\to Ma{t}^{\left(2,2\right)}$ given by $f\left(x,y\right)=\left[\begin{array}{cc}x& 2y\\ x+y& x\end{array}\right]$We calculate image of canonical basis $f\left(1,0\right)=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $f\left(0,1\right)=\left[\begin{array}{cc}0& 2\\ 1& 0\end{array}\right]$Now the problematic part, ever since when caltulating matrix from vectors ${R}^{n}\to {R}^{m}$ the approach is to transpose images of standard base $\left(f\left(1,0,...,0{\right)}^{T}|f\left(0,1,...,0{\right)}^{T}|f\left(0,0,...,1{\right)}^{T}\right)$. We can solve the $R\to Mat$ problem by using $\left({A}^{T}|{B}^{T}|{C}^{T}...\right)$, of course, but is there any way how to shrink the vector so we can succeed something like $\left({A}^{T}|{B}^{T}|{C}^{T}...\right)$?

vballa15ei 2022-09-08

Let $T:V\to W$ be a linear transformation of two finite dimensional vector spaces $V$,$W$ (both over a field $F$).My assignment is to show that there exists a basis $B$ in $V$ and a basis $C$ in $W$ such that the transformation $F$ with respect to the bases $B$ and $C$ actually does have $A$ as a transformation matrix.Since $V$,$W$ are vector spaces of course they have bases and I can do a transformation with respect to the bases from $V$ to $W$. But how do I show that (some specific?) bases exist such that the transformation with respect to these bases have a given a $A$ as a transformation matrix?Assume $A$ is transformation matrix of $T$. Then we take one vector in $V$ expressed in (some basis) $B$ and when we transform it to some vector in $W$, expressed in (some basis) 𝐶. So $A$ does at least change the basis of a vector, but I since $V$ and $W$ might be of different dimension I don't really know where to go from here. Am I one the wrong track?

Spactapsula2l 2022-09-07

Linear transformation, T, such that:T:${M}_{22}$$\left(\left[\begin{array}{cc}{x}_{11}& {x}_{12}\\ {x}_{21}& {x}_{22}\end{array}\right]\right)=\left[\begin{array}{cc}{x}_{12}-5{x}_{21}-{x}_{22}& -{x}_{11}-2{x}_{12}+3{x}_{21}+4{x}_{22}\\ -3{x}_{21}& -{x}_{11}-{x}_{12}+{x}_{21}+3{x}_{22}\end{array}\right]$What is the matrix that represents this ${M}_{22}$->${M}_{22}$ transformation?Is it:$\left[\begin{array}{cccc}0& 1& -5& -1\\ -1& -2& 3& 4\\ 0& 0& -3& 0\\ -1& -1& 1& 3\end{array}\right]$If so, how could this be multiplied by a 2x2 matrix to give another 2x2 matrix. (2x2 matrices cannot multiply with 4x4 matrices).

Linear algebraOpen question
decorraj9 2022-08-31

Find the inverse, if it exists, for the given matrix$\left[\begin{array}{cc}3& 2\\ 2& 5\end{array}\right]$

Ashlynn Hale 2022-08-10

An augmented matrix of a system of equations has been transformed by row operations into the 3 equations in 3 variables matrix below. State the solution to this system as (x,y,z) Enter your answer as an ordered triple within parenthesis separated by commas (x,y,z). Avoid entering spaces and decimal points Use fractions if relevant.$\left[\begin{array}{ccccc}1& 0& 0& ⋮& -3\\ 0& 1& 0& ⋮& \frac{2}{3}\\ 0& 0& 1& ⋮& \frac{1}{4}\end{array}\right]$

cofak48 2022-08-04

Let $A=\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-7& 9\\ 1& -9\end{array}\right]$Find 3A-2B.

Gorlandint 2022-08-02

Perform the indicated matrix operation.$\left[\begin{array}{ccc}2x+y& x-2y& 4x\\ 3x& 3y& x+y\end{array}\right]+\left[\begin{array}{ccc}5x& 8y& 3x+y\\ 5x+5y& x& 2x\end{array}\right]$

Spoorthi 2022-07-31