Recent questions in Matrix transformations

Linear algebraAnswered question

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?

Linear algebraAnswered question

Julia Chang 2022-09-25

Give the standard matrix of the linear transformation that first sends (x, y, z) to (y, y, z), and rotates this vector 90 degrees counterclockwise about the origin in the x = y plane. Find standard matrix of linear transformation.

Linear algebraAnswered question

Colten Andrade 2022-09-21

Given a matrix $Y\in {\mathbb{R}}^{m\times n}$. Find a transformation matrix $\mathrm{\Theta}\in {\mathbb{R}}^{n\times p}$ such that

$$\frac{1}{m}}{\mathrm{\Theta}}^{T}{Y}^{T}Y\mathrm{\Theta}={I}_{p\times p},$$

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.

$$\frac{1}{m}}{\mathrm{\Theta}}^{T}{Y}^{T}Y\mathrm{\Theta}={I}_{p\times p},$$

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.

Linear algebraAnswered question

Celinamg8 2022-09-20

Vector, $u:=[{u}_{1},\dots ,{u}_{n}{]}^{\mathrm{T}}$. I am trying to find a coordinate transformation matrix, $Q\in {\mathbb{R}}^{n\times n}$, which is nonsingular, satisfying:

$$\begin{array}{r}\left[\begin{array}{c}0\\ \vdots \\ 0\\ ||u||\end{array}\right]=Qu.\end{array}$$

$$\begin{array}{r}\left[\begin{array}{c}0\\ \vdots \\ 0\\ ||u||\end{array}\right]=Qu.\end{array}$$

Linear algebraAnswered question

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?

Linear algebraAnswered question

nar6jetaime86 2022-09-13

Findthe Matrix T of the following linear transformation

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

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

Linear algebraAnswered question

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?

Linear algebraAnswered question

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 $||T||=inf\{M\ge 0:||T(x)||\le M||x||\text{}\mathrm{\forall}x\in {\mathbb{K}}^{n}\}$ 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:

$$||T||=max\{\sum _{i=1}^{m}|{a}_{ij}|,1\le j\le n\}=:||A|{|}_{1}$$

where $A$ is the transformation matrix of $T$ and ${a}_{ij}$ it's entry in the $i$-th row and $j$-the column.

Let $||T||=inf\{M\ge 0:||T(x)||\le M||x||\text{}\mathrm{\forall}x\in {\mathbb{K}}^{n}\}$ 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:

$$||T||=max\{\sum _{i=1}^{m}|{a}_{ij}|,1\le j\le n\}=:||A|{|}_{1}$$

where $A$ is the transformation matrix of $T$ and ${a}_{ij}$ it's entry in the $i$-th row and $j$-the column.

Linear algebraAnswered question

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:

$$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 0& 1& 0& 1\end{array}\right]$$

$$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

To the following:

$$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 0& 1& 0& 1\end{array}\right]$$

Linear algebraAnswered question

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.

Linear algebraAnswered question

moidu13x8 2022-09-09

How to find a matrix of linear transformation $f:{R}^{n}\to Ma{t}^{(n,n)}$.

Let's say we do $f:{R}^{2}\to Ma{t}^{(2,2)}$ given by $f(x,y)=\left[\begin{array}{cc}x& 2y\\ x+y& x\end{array}\right]$

We calculate image of canonical basis $f(1,0)=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $f(0,1)=\left[\begin{array}{cc}0& 2\\ 1& 0\end{array}\right]$

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 $(f(1,0,...,0{)}^{T}|f(0,1,...,0{)}^{T}|f(0,0,...,1{)}^{T})$. We can solve the $R\to Mat$ problem by using $({A}^{T}|{B}^{T}|{C}^{T}...)$, of course, but is there any way how to shrink the vector so we can succeed something like $({A}^{T}|{B}^{T}|{C}^{T}...)$?

Let's say we do $f:{R}^{2}\to Ma{t}^{(2,2)}$ given by $f(x,y)=\left[\begin{array}{cc}x& 2y\\ x+y& x\end{array}\right]$

We calculate image of canonical basis $f(1,0)=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $f(0,1)=\left[\begin{array}{cc}0& 2\\ 1& 0\end{array}\right]$

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 $(f(1,0,...,0{)}^{T}|f(0,1,...,0{)}^{T}|f(0,0,...,1{)}^{T})$. We can solve the $R\to Mat$ problem by using $({A}^{T}|{B}^{T}|{C}^{T}...)$, of course, but is there any way how to shrink the vector so we can succeed something like $({A}^{T}|{B}^{T}|{C}^{T}...)$?

Linear algebraAnswered question

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?

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?

Linear algebraAnswered question

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).

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

$\left[\begin{array}{cc}3& 2\\ 2& 5\end{array}\right]$

$\left[\begin{array}{cc}3& 2\\ 2& 5\end{array}\right]$

Linear algebraAnswered question

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.

$$\left[\begin{array}{ccccc}1& 0& 0& \vdots & -3\\ 0& 1& 0& \vdots & \frac{2}{3}\\ 0& 0& 1& \vdots & \frac{1}{4}\end{array}\right]$$

Linear algebraAnswered question

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.

Find 3A-2B.

Linear algebraAnswered question

Gorlandint 2022-08-02

Perform the indicated matrix operation.

$\left[\begin{array}{ccc}2x+y& x-2y& 4x\\ 3x& 3y& x+y\end{array}\right]+\left[\begin{array}{ccc}5x& 8y& 3x+y\\ 5x+5y& x& 2x\end{array}\right]$

$\left[\begin{array}{ccc}2x+y& x-2y& 4x\\ 3x& 3y& x+y\end{array}\right]+\left[\begin{array}{ccc}5x& 8y& 3x+y\\ 5x+5y& x& 2x\end{array}\right]$

Linear algebraAnswered question

Spoorthi 2022-07-31

Let T be the linear transformation whose standard matrix is given below. A=[7 5 4 -9 ,10 6 16 -4 ,12 8 12 7, -8 -6 -2 5] a) Decide if T is a one-to-one mapping.

b)Decide if Rn is mapped onto Rm

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.