Recent questions in Matrix transformations

Linear algebraAnswered question

Faith Welch 2022-07-26

Find the production matrix for the following input-output and demand matrices using the open model.

$A=\left[\begin{array}{cc}-0.1& 0.2\\ 0.55& 0.4\end{array}\right]$

$D=\left[\begin{array}{c}3\\ 4\end{array}\right]$

$A=\left[\begin{array}{cc}-0.1& 0.2\\ 0.55& 0.4\end{array}\right]$

$D=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Linear algebraAnswered question

Jaylene Hunter 2022-07-26

Solve the following system of equations by using the inverse of the coefficient matrix A.

(AX=B), x+5y=-10, -2x+7y=-31

(AX=B), x+5y=-10, -2x+7y=-31

Linear algebraAnswered question

Kade Reese 2022-07-25

If A is an n x n matrix , where are the entries on the main diagonal of A-A^T? Justify yoyr answer.

Linear algebraAnswered question

Usman Zahid2022-07-24

a5=0 and a15=4 what is the sum of the first 15 terms of that arithmetic sequence

Linear algebraAnswered question

Brock Byrd 2022-07-16

Let $T:{M}_{2x2}\to {\mathbb{R}}^{3}$ have matrix $[T{]}_{B,A}=\left[\begin{array}{cccc}1& 2& 0& 1\\ 0& 1& 1& 0\\ 1& 1& -1& -1\end{array}\right]$ relative to $A=\{\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 3\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 5& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 6\end{array}\right]\}$ and $\beta =\{(1,1,1),(1,2,3),(1,4,9)\}$. Find the matrix of T relative to the bases ${A}^{\prime}=\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 4\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 2& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 7\end{array}\right]\}$ and ${\beta}^{\prime}=\{(1,1,1),(1,0,0),(1,1,0)\}$

Linear algebraAnswered question

Joshua Foley 2022-07-16

representative matrix of a linear transformation

given a linear transformation: $T:{M}_{n}(\mathbb{C})\to {M}_{n}(\mathbb{C})$, $T(A)=A-2{A}^{T}$, what is the

given a linear transformation: $T:{M}_{n}(\mathbb{C})\to {M}_{n}(\mathbb{C})$, $T(A)=A-2{A}^{T}$, what is the

Linear algebraAnswered question

Rebecca Villa 2022-07-15

Let V be a 3 dimensional vector space over a field F and fix $({\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},{\mathbf{v}}_{\mathbf{3}})$ as a basis. Consider a linear transformation $T:V\to V$. Then we have

$T({\mathbf{v}}_{\mathbf{1}})={a}_{11}{\mathbf{v}}_{\mathbf{1}}+{a}_{21}{\mathbf{v}}_{\mathbf{2}}+{a}_{31}{\mathbf{v}}_{\mathbf{3}}$

$T({\mathbf{v}}_{\mathbf{2}})={a}_{12}{\mathbf{v}}_{\mathbf{1}}+{a}_{22}{\mathbf{v}}_{\mathbf{2}}+{a}_{32}{\mathbf{v}}_{\mathbf{3}}$

$T({\mathbf{v}}_{\mathbf{3}})={a}_{13}{\mathbf{v}}_{\mathbf{1}}+{a}_{23}{\mathbf{v}}_{\mathbf{2}}+{a}_{33}{\mathbf{v}}_{\mathbf{3}}$

So that we can identify T by the matrix

$\left(\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right)$

But then when I read several linear algebra book, it said if $T({\mathbf{v}}_{\mathbf{i}})=\sum _{j}{a}_{ij}{\mathbf{v}}_{\mathbf{j}}$ , then we can identify T by the matrix $({a}_{ij})$. My problem is: isn't the matrix is $({a}_{ji})$ instead of $({a}_{ij})$?

$T({\mathbf{v}}_{\mathbf{1}})={a}_{11}{\mathbf{v}}_{\mathbf{1}}+{a}_{21}{\mathbf{v}}_{\mathbf{2}}+{a}_{31}{\mathbf{v}}_{\mathbf{3}}$

$T({\mathbf{v}}_{\mathbf{2}})={a}_{12}{\mathbf{v}}_{\mathbf{1}}+{a}_{22}{\mathbf{v}}_{\mathbf{2}}+{a}_{32}{\mathbf{v}}_{\mathbf{3}}$

$T({\mathbf{v}}_{\mathbf{3}})={a}_{13}{\mathbf{v}}_{\mathbf{1}}+{a}_{23}{\mathbf{v}}_{\mathbf{2}}+{a}_{33}{\mathbf{v}}_{\mathbf{3}}$

So that we can identify T by the matrix

$\left(\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right)$

But then when I read several linear algebra book, it said if $T({\mathbf{v}}_{\mathbf{i}})=\sum _{j}{a}_{ij}{\mathbf{v}}_{\mathbf{j}}$ , then we can identify T by the matrix $({a}_{ij})$. My problem is: isn't the matrix is $({a}_{ji})$ instead of $({a}_{ij})$?

Linear algebraAnswered question

Addison Trujillo 2022-07-15

Let A be the matrix below and define a transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ by $T(U)=AU.$. For each of the vectors $B$ below, find a vector $U$ such that $T$ maps $U$ to $B$, if possible. Otherwise state that there is no such $U$.

$\left(\begin{array}{ccc}1& -3& 2\\ 2& -4& 4\\ 3& -8& 6\end{array}\right)=A$

a) $\left(\begin{array}{c}4\\ 6\\ 11\end{array}\right)=B$

b) $\left(\begin{array}{c}-3\\ -2\\ -7\end{array}\right)=B$

$\left(\begin{array}{ccc}1& -3& 2\\ 2& -4& 4\\ 3& -8& 6\end{array}\right)=A$

a) $\left(\begin{array}{c}4\\ 6\\ 11\end{array}\right)=B$

b) $\left(\begin{array}{c}-3\\ -2\\ -7\end{array}\right)=B$

Linear algebraAnswered question

uplakanimkk 2022-07-14

Suppose $T$ is a transformation from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$. Find the matrix $A$ that induces $T$ if $T$ is the (counter-clockwise) rotation by $\frac{3}{4}}\pi $.

how to begin to find a matrix that is 2x2 for this question.

how to begin to find a matrix that is 2x2 for this question.

Linear algebraAnswered question

Willow Pratt 2022-07-12

Let $A$ be the set of all $n\times n$ symmetric real matirix and $f\in C(\mathbb{R},A)$. Then whether there is a $g\in C(\mathbb{R},O(n))$ such that for all $t\in \mathbb{R}$, $g(t{)}^{-1}f(t)g(t)$ is a diagonal matrix?

Linear algebraAnswered question

Jamison Rios 2022-07-12

Can't figure out this transformation matrix

1) The positive z axis normalized as Vector(0,0,1) has to map to an arbitrary direction vector in the new coordinate system Vector(a,b,c)

2) The origin in the original coordinate system has to map to an arbitrary position P in the new coordinate system.

3) This might be redundant but the positive Y axis has to map to a specific direction vector(d,e,f) which is perpendicular to Vector(a,b,c) from before.

So my question is twofold: 1) How would I go about constructing this transformation matrix and 2) Is this enough data to ensure that any arbitrary vector in coordinate system 1 will be accurately transformed in coordinate system 2?

1) The positive z axis normalized as Vector(0,0,1) has to map to an arbitrary direction vector in the new coordinate system Vector(a,b,c)

2) The origin in the original coordinate system has to map to an arbitrary position P in the new coordinate system.

3) This might be redundant but the positive Y axis has to map to a specific direction vector(d,e,f) which is perpendicular to Vector(a,b,c) from before.

So my question is twofold: 1) How would I go about constructing this transformation matrix and 2) Is this enough data to ensure that any arbitrary vector in coordinate system 1 will be accurately transformed in coordinate system 2?

Linear algebraAnswered question

ntaraxq 2022-07-12

How can one prove the following identity of the cross product?

$(Ma)\times (Mb)=det(M)({M}^{\mathrm{T}}{)}^{-1}(a\times b)$

$a$ and $b$ are 3-vectors, and $M$ is an invertible real $3\times 3$ matrix.

$(Ma)\times (Mb)=det(M)({M}^{\mathrm{T}}{)}^{-1}(a\times b)$

$a$ and $b$ are 3-vectors, and $M$ is an invertible real $3\times 3$ matrix.

Linear algebraAnswered question

mistergoneo7 2022-07-10

I'd like to be able to enter a vector or matrix, see it in 2-space or 3-space, enter a transformation vector or matrix, and see the result. For example, enter a 3x3 matrix, see the parallelepiped it represents, enter a rotation matrix, see the rotated parallelepiped.

Linear algebraAnswered question

Kristen Stokes 2022-07-10

Let $M$ be a transformation matrix $B\to {B}^{\prime}$

discovered that ${M}^{-1}$ is the opposite transformation.

What makes it true?

discovered that ${M}^{-1}$ is the opposite transformation.

What makes it true?

Linear algebraAnswered question

ntaraxq 2022-07-10

Consider the transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{2}$ defined by:

$T(x)=T\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=(2{x}_{1}+{x}_{3})\left(\begin{array}{c}1\\ 2\end{array}\right)+({x}_{2}-3{x}_{3})\left(\begin{array}{c}-1\\ 1\end{array}\right)$

1a) determine the matrix of the above transformation

1b) determine the reduced row echelon form of the matrix found in 1a

1c) based on your answer to part 1b, is the transformation T onto?

1d) based on your answer to part 1b, is the transformation T one-to-one?

1e) based on your answer to part 1b, determine the set of vectors $x$ in ${\mathbb{R}}^{3}$ for which $T(x)=0$. Write your answer in parametric vector form

$T(x)=T\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=(2{x}_{1}+{x}_{3})\left(\begin{array}{c}1\\ 2\end{array}\right)+({x}_{2}-3{x}_{3})\left(\begin{array}{c}-1\\ 1\end{array}\right)$

1a) determine the matrix of the above transformation

1b) determine the reduced row echelon form of the matrix found in 1a

1c) based on your answer to part 1b, is the transformation T onto?

1d) based on your answer to part 1b, is the transformation T one-to-one?

1e) based on your answer to part 1b, determine the set of vectors $x$ in ${\mathbb{R}}^{3}$ for which $T(x)=0$. Write your answer in parametric vector form

Linear algebraAnswered question

prirodnogbk 2022-07-09

Can't figure out this transformation matrix

1) The positive z axis normalized as Vector(0,0,1) has to map to an arbitrary direction vector in the new coordinate system Vector(a,b,c)

2) The origin in the original coordinate system has to map to an arbitrary position P in the new coordinate system.

3) This might be redundant but the positive Y axis has to map to a specific direction vector(d,e,f) which is perpendicular to Vector(a,b,c) from before.

So my question is twofold: 1) How would I go about constructing this transformation matrix and 2) Is this enough data to ensure that any arbitrary vector in coordinate system 1 will be accurately transformed in coordinate system 2?

1) The positive z axis normalized as Vector(0,0,1) has to map to an arbitrary direction vector in the new coordinate system Vector(a,b,c)

2) The origin in the original coordinate system has to map to an arbitrary position P in the new coordinate system.

3) This might be redundant but the positive Y axis has to map to a specific direction vector(d,e,f) which is perpendicular to Vector(a,b,c) from before.

So my question is twofold: 1) How would I go about constructing this transformation matrix and 2) Is this enough data to ensure that any arbitrary vector in coordinate system 1 will be accurately transformed in coordinate system 2?

Linear algebraAnswered question

uplakanimkk 2022-07-09

Find the transformation matrix

Let $B=\{2x,3x+{x}^{2},-1\},{B}^{\prime}=\{1,1+x,1+x+{x}^{2}\}$

Need to find the transformation matrix from $B$ to ${B}^{\prime}$.

I know that:

$(a{x}^{2}+bx+c{)}_{B}=(\frac{b-3c}{2},c,-a)$

$(a{x}^{2}+bx+c{)}_{{B}^{\prime}}=(a-b,b-c,c)$

How to proceed using this info in order to find the transformation matrix?

Let $B=\{2x,3x+{x}^{2},-1\},{B}^{\prime}=\{1,1+x,1+x+{x}^{2}\}$

Need to find the transformation matrix from $B$ to ${B}^{\prime}$.

I know that:

$(a{x}^{2}+bx+c{)}_{B}=(\frac{b-3c}{2},c,-a)$

$(a{x}^{2}+bx+c{)}_{{B}^{\prime}}=(a-b,b-c,c)$

How to proceed using this info in order to find the transformation matrix?

Linear algebraAnswered question

kramberol 2022-07-08

Let $\beta $ and ${\beta}^{\prime}$ be bases for the finite dimensional vector space $V$ of dimension n over the field $\mathbb{F}$, and let $Q=[{I}_{V}{]}_{{\beta}^{\prime}}^{\beta}$, where ${I}_{V}$ is the identity operator on $V$. I just recently proved that $Q[x{]}_{{\beta}^{\prime}}=[x{]}_{\beta}$ for every $x\in V$ (twas rather simple), which suggests the title of "basis transformation" matrix or "coordinate transformation" matrix for the matrix $Q$. I am now wondering whether the converse holds.

Let ${V}^{\prime}$ denote $V$ with its elements written with respect to the basis ${\beta}^{\prime}$, and suppose that $Q\in {M}_{n}(\mathbb{F})$ is such that $Q[x{]}_{{\beta}^{\prime}}=[x{]}_{\beta}$ for every $x$. Since $\mathcal{L}({V}^{\prime}V)$ is isomorphic to the matrix algebra ${M}_{n}(\mathbb{F})$ by sending a linear operator to its matrix representation, given $Q$ there exists a linear operator $T\in \mathcal{L}({V}^{\prime},V)$ such that $Q=[T{]}_{{\beta}^{\prime}}^{\beta}$. Hence $[T{]}_{{\beta}^{\prime}}^{\beta}[x{]}_{{\beta}^{\prime}}=[x{]}_{\beta}$ or $[T(x){]}_{\beta}=[x{]}_{\beta}$...

I want to say that this implies $T={I}_{V}$, but I can't clearly see what lemma I need in order to make that conclusion.

Let ${V}^{\prime}$ denote $V$ with its elements written with respect to the basis ${\beta}^{\prime}$, and suppose that $Q\in {M}_{n}(\mathbb{F})$ is such that $Q[x{]}_{{\beta}^{\prime}}=[x{]}_{\beta}$ for every $x$. Since $\mathcal{L}({V}^{\prime}V)$ is isomorphic to the matrix algebra ${M}_{n}(\mathbb{F})$ by sending a linear operator to its matrix representation, given $Q$ there exists a linear operator $T\in \mathcal{L}({V}^{\prime},V)$ such that $Q=[T{]}_{{\beta}^{\prime}}^{\beta}$. Hence $[T{]}_{{\beta}^{\prime}}^{\beta}[x{]}_{{\beta}^{\prime}}=[x{]}_{\beta}$ or $[T(x){]}_{\beta}=[x{]}_{\beta}$...

I want to say that this implies $T={I}_{V}$, but I can't clearly see what lemma I need in order to make that conclusion.

Linear algebraAnswered question

Audrina Jackson 2022-07-08

Consider the 4-by-4 matrix $\mathit{M}={\mathit{M}}_{0}+{\mathit{M}}_{1}$, where

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

Linear algebraAnswered question

orlovskihmw 2022-07-07

The transformation $\mathbf{\text{T}}$ maps points $(x,y)$ of the plane into image points $({x}^{\prime},{y}^{\prime})$ such that

$\begin{array}{rl}{x}^{\prime}& =4x+2y+14\\ {y}^{\prime}& =2x+7y+42\end{array}$

Find the coordinates of the invariant point of $\mathbf{\text{T}}$. Hence express $\mathbf{\text{T}}$ in the form

$\left(\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}+k\end{array}\right)=\mathbf{\text{A}}\left(\begin{array}{c}x\\ y+k\end{array}\right)$

where $k$ is a positive integer and $\mathbf{\text{A}}$ is a $2\times 2$ matrix.

$\begin{array}{rl}{x}^{\prime}& =4x+2y+14\\ {y}^{\prime}& =2x+7y+42\end{array}$

Find the coordinates of the invariant point of $\mathbf{\text{T}}$. Hence express $\mathbf{\text{T}}$ in the form

$\left(\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}+k\end{array}\right)=\mathbf{\text{A}}\left(\begin{array}{c}x\\ y+k\end{array}\right)$

where $k$ is a positive integer and $\mathbf{\text{A}}$ is a $2\times 2$ matrix.

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.