# Matrix transformation questions and answers

Recent questions in Matrix transformations
fatihahjohari18 2023-01-03

### 4 ^ x - 2 ^ (x + 1) - 3 = 0

oxidricasbt7 2022-12-20

### Eigenvalues of a $2×2$ matrix A such that ${A}^{2}=I$I have no idea where to begin.I know there are a few matrices that support this claim, will they all have the same eigenvalues?

Jayden Davidson 2022-12-18

### Can someone explain why a row replacement operation does not change the determinant of a matrix?

criminauiac 2022-12-14

### How do you evaluate a matrix.

e3r2a1cakCh7 2022-12-03

### The matrix equation is not solved correctly. Expain the mistake and find the correct solution. Assume that the indicated inverses exist. $XA=B,X={A}^{-1}B$

atgnybo4fq 2022-11-12

### Determining the Rank of a Matrix$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$

Hugo Stokes 2022-10-18

### Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?

Paloma Sanford 2022-10-13

### Let $A=4×4$ matrix: $\left[\begin{array}{cccc}3& 2& 10& -6\\ 1& 0& 2& -4\\ 0& 1& 2& 3\\ 1& 4& 10& 8\end{array}\right]$, let $b=4×1$ matrix: $\left[\begin{array}{c}-1\\ 3\\ -1\\ 4\end{array}\right]$Is $b$ in the range of linear transformation $x\to Ax$?

Krish Schmitt 2022-09-30

### Suppose $V$ is a $n$-dimensional linear vector space. $\left\{{s}_{1},{s}_{2},...,{s}_{n}\right\}$ and $\left\{{e}_{1},{e}_{2},...,{e}_{n}\right\}$ are two sets of orthonormal basis with basis transformation matrix $U$ such that ${e}_{i}=\sum _{j}{U}_{ij}{s}_{j}$.Now consider the ${n}^{2}$ dim vector space $V⨂V$ (kronecker product) with equivalent basis sets $\left\{{s}_{1}{s}_{1},{s}_{1}{s}_{2},...,{s}_{n}{s}_{n}\right\}$ and $\left\{{e}_{1}{e}_{1},{e}_{1}{e}_{2},...,{e}_{n}{e}_{n}\right\}$. Now can we find the basis transformation matrix for this in terms of U?

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 zubair 2022-09-14

### 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?

ahmed zubair 2022-09-14

### 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.

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.