Recent questions in Matrix transformations

Linear algebraAnswered question

Ciara Mcdaniel 2022-07-01

Find the transformation matrix R that relates the (orthonormal ) standard basis of ${\mathbb{C}}^{3}$ to the orthonormal basis obtained from the following vectors via the Gram Schmidt process:

a1> = $\left(\begin{array}{c}1\\ i\\ 0\end{array}\right)$

a2> = $\left(\begin{array}{c}0\\ 1\\ -i\end{array}\right)$

a3> = $\left(\begin{array}{c}i\\ 0\\ -1\end{array}\right)$

a1> = $\left(\begin{array}{c}1\\ i\\ 0\end{array}\right)$

a2> = $\left(\begin{array}{c}0\\ 1\\ -i\end{array}\right)$

a3> = $\left(\begin{array}{c}i\\ 0\\ -1\end{array}\right)$

Linear algebraAnswered question

hornejada1c 2022-07-01

Finding inverse matrix ${A}^{-1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A.

$A=\left(\begin{array}{cccc}1& 3& 9& 27\\ 3& 3& 9& 27\\ 9& 9& 9& 27\\ 27& 27& 27& 27\end{array}\right)$

$B=\left(\begin{array}{cccc}1& 3& 9& 27\\ 1& 1& 3& 9\\ 1& 1& 1& 3\\ 1& 1& 1& 1\end{array}\right)$

I found that that ${B}^{-1}$ is

${B}^{-1}=\left(\begin{array}{cccc}-\frac{1}{2}& \frac{3}{2}& 0& 0\\ \frac{1}{2}& -2& \frac{3}{2}& 0\\ 0& \frac{1}{2}& -2& \frac{3}{2}\\ 0& 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right)$

I don't know how to continue. What rule do I use to find ${A}^{-1}$?

$A=\left(\begin{array}{cccc}1& 3& 9& 27\\ 3& 3& 9& 27\\ 9& 9& 9& 27\\ 27& 27& 27& 27\end{array}\right)$

$B=\left(\begin{array}{cccc}1& 3& 9& 27\\ 1& 1& 3& 9\\ 1& 1& 1& 3\\ 1& 1& 1& 1\end{array}\right)$

I found that that ${B}^{-1}$ is

${B}^{-1}=\left(\begin{array}{cccc}-\frac{1}{2}& \frac{3}{2}& 0& 0\\ \frac{1}{2}& -2& \frac{3}{2}& 0\\ 0& \frac{1}{2}& -2& \frac{3}{2}\\ 0& 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right)$

I don't know how to continue. What rule do I use to find ${A}^{-1}$?

Linear algebraAnswered question

Cristopher Knox 2022-07-01

Prove that pre-multiplying a matrix ${A}_{m}$ by the elementary matrix obtained with any matrix elementary line transformation ${I}_{m}\underset{{l}_{1}\leftrightarrow {l}_{2}}{\u27f6}E$ is the same as applying said elementary line transformation on the matrix ${A}_{m}$

Linear algebraAnswered question

Kristen Stokes 2022-07-01

$T(p(x))={\int}_{0}^{1}p(x)dx.$

(a) Show $T$ is a linear transformation.

(b) Compute $\mathcal{N}(T).$ Is $T$ one-to-one?

(c) Show that $T$ is onto.

(d) Let $B$ be the standard basis for ${\mathcal{P}}_{2}$ and let ${B}^{\mathrm{\prime}}=\{1\}$ be a basis for $\mathbb{R}$. Find $[T{]}_{B}^{{B}^{\mathrm{\prime}}}$.

(e) Use the matrix found in part (d) to compute $T(-{x}^{2}-3x+2)$

(a) Show $T$ is a linear transformation.

(b) Compute $\mathcal{N}(T).$ Is $T$ one-to-one?

(c) Show that $T$ is onto.

(d) Let $B$ be the standard basis for ${\mathcal{P}}_{2}$ and let ${B}^{\mathrm{\prime}}=\{1\}$ be a basis for $\mathbb{R}$. Find $[T{]}_{B}^{{B}^{\mathrm{\prime}}}$.

(e) Use the matrix found in part (d) to compute $T(-{x}^{2}-3x+2)$

Linear algebraAnswered question

spockmonkey40 2022-06-30

Find the matrix A for the linear transformation T relative to the bases $B=\{1,x,{x}^{2}\}$ and $B\prime =\{1,x,{x}^{2},{x}^{3}\}$ such that $T\left(\overrightarrow{x}\right)=A\overrightarrow{x}$

Linear algebraAnswered question

kokoszzm 2022-06-30

Find the transformation matrix:

$F:{\mathbb{R}}_{3}[x]\text{}\mathbb{]}\to {\mathbb{R}}_{3}[x]$

$F(v)=\frac{{d}^{2}v}{d{v}^{2}}$

Basis: $1,x,{x}^{2},{x}^{3}$ and ${\mathbb{R}}_{3}[x]$ - the set of all third degree polynomials of variable $x$ over $\mathbb{R}$ Assume that all coefficients of the polynomials are $1$

The first thing that springs to my mind is to calculate this derivative by hand, and so we got

$\frac{{d}^{2}y}{d{y}^{2}}=2+6x$

Now, we need to put these values - $2$ and $6$ in such a matrix that - when multiplied by the basis vector -will give us $2+6x$ But there are many ways I can think of, for example

$\left[\begin{array}{cccc}0& 0& 2& 0\\ 0& 0& 0& 6\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

Or maybe

$\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 6& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

Because both of them, when multiplied by$\left[\begin{array}{c}1\\ x\\ {x}^{2}\\ {x}^{3}\end{array}\right]$

Will give the correct answer. Thus, what is the correct way to solve this?

$F:{\mathbb{R}}_{3}[x]\text{}\mathbb{]}\to {\mathbb{R}}_{3}[x]$

$F(v)=\frac{{d}^{2}v}{d{v}^{2}}$

Basis: $1,x,{x}^{2},{x}^{3}$ and ${\mathbb{R}}_{3}[x]$ - the set of all third degree polynomials of variable $x$ over $\mathbb{R}$ Assume that all coefficients of the polynomials are $1$

The first thing that springs to my mind is to calculate this derivative by hand, and so we got

$\frac{{d}^{2}y}{d{y}^{2}}=2+6x$

Now, we need to put these values - $2$ and $6$ in such a matrix that - when multiplied by the basis vector -will give us $2+6x$ But there are many ways I can think of, for example

$\left[\begin{array}{cccc}0& 0& 2& 0\\ 0& 0& 0& 6\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

Or maybe

$\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 6& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

Because both of them, when multiplied by$\left[\begin{array}{c}1\\ x\\ {x}^{2}\\ {x}^{3}\end{array}\right]$

Will give the correct answer. Thus, what is the correct way to solve this?

Linear algebraAnswered question

Jameson Lucero 2022-06-29

Consider in ${\mathbb{R}}^{2}$ the set of points satisfying the equation (you can use matlab for this). How are these points transformed by the following matrix:

$\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]$

[Show the transformed set and plot it – you can use matlab again]

I could be mistaken but would the matrix be

$\left[\begin{array}{cc}2& -1-2\end{array}\right]?$

After that I am very stuck. Thanks

$\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]$

[Show the transformed set and plot it – you can use matlab again]

I could be mistaken but would the matrix be

$\left[\begin{array}{cc}2& -1-2\end{array}\right]?$

After that I am very stuck. Thanks

Linear algebraAnswered question

veneciasp 2022-06-29

If the transformation is from ${\mathbb{R}}^{3}\to \mathbb{R}$ is

$T\{a,b,c\}={\int}_{0}^{\pi}2a{e}^{t}+2b\mathrm{sin}(t)+3c\mathrm{cos}(t)\phantom{\rule{thinmathspace}{0ex}}dt$

How to find the standard matrix?

$T\{a,b,c\}={\int}_{0}^{\pi}2a{e}^{t}+2b\mathrm{sin}(t)+3c\mathrm{cos}(t)\phantom{\rule{thinmathspace}{0ex}}dt$

How to find the standard matrix?

Linear algebraAnswered question

bandikizaui 2022-06-29

In an affine transformation $x\mapsto Ax+b$, $b$ represents the translation; but what does the matrix A represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term represent?

Linear algebraAnswered question

Devin Anderson 2022-06-29

If I take a particular number, for example 21, and write down all possible matrices whose determinant is 21, will all these matrices represent the same transformation with different system of bases? Consider all endomorphisms.

Linear algebraAnswered question

Makayla Boyd 2022-06-28

Transformation matrix for matrix indices to cartesian coordinates

(1,1) (1,2) (1,3)(2,1) (2,2) (2,3)(3,1) (3,2) (3,3)

This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation would be:

(-1,1) (0,1) (1,1)(-1,0) (0,0) (1,0)(-1,1) (0,-1) (1,-1)

Say, I have any square matrix with dimension-N, where N is odd. I need a generic transformation matrix such that I can get a vector as cartesian coordinates from matrix indices. Does such a function already exist? How should I go ahead in implementing this?

(1,1) (1,2) (1,3)(2,1) (2,2) (2,3)(3,1) (3,2) (3,3)

This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation would be:

(-1,1) (0,1) (1,1)(-1,0) (0,0) (1,0)(-1,1) (0,-1) (1,-1)

Say, I have any square matrix with dimension-N, where N is odd. I need a generic transformation matrix such that I can get a vector as cartesian coordinates from matrix indices. Does such a function already exist? How should I go ahead in implementing this?

Linear algebraAnswered question

Dania Mueller 2022-06-27

Let $v\in {T}_{2}(V)$ be a bilinear form over finite space V. Let T be a Linear transformation $V\to V$. We define: ${v}_{T}(x,y)=v(T(x),y)$

Assuming $v$ is nondegenerate, let us have another bilinear form $\xi \in {T}_{2}(V)$. Prove that there exists exactly one transformation $T$ so $\xi ={v}_{T}$.

Assuming $v$ is nondegenerate, let us have another bilinear form $\xi \in {T}_{2}(V)$. Prove that there exists exactly one transformation $T$ so $\xi ={v}_{T}$.

Linear algebraAnswered question

Poftethef9t 2022-06-27

Let's say we define the orientation of the ellipsoid from its major axis (the largest axis of the ellipsoid). Assuming the 3 axes of the ellipsoid to be on the three coordinates with lengths of $a$, $b$ and $c$ along each axis (with $a\ge b$ and $a\ge c$), then only a single affine transformation (to a sphere) is necessary:

$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]$

Now, if we first rotate the major axis by $\theta $ from the first axis towards the second axis, and then rotate it by $\varphi $ from the (rotated) first axis towards the third axis, the combined affine transformation becomes:

$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\varphi & 0& -sin\varphi & 0\\ 0& 1& 0& 0\\ sin\varphi & 0& cos\varphi & 0\\ 0& 0& 0& 1\end{array}\right]$

Is the multiplied matrix (from left to right) the correct affine transformation that must go into the equations in the links above?

$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]$

Now, if we first rotate the major axis by $\theta $ from the first axis towards the second axis, and then rotate it by $\varphi $ from the (rotated) first axis towards the third axis, the combined affine transformation becomes:

$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\varphi & 0& -sin\varphi & 0\\ 0& 1& 0& 0\\ sin\varphi & 0& cos\varphi & 0\\ 0& 0& 0& 1\end{array}\right]$

Is the multiplied matrix (from left to right) the correct affine transformation that must go into the equations in the links above?

Linear algebraAnswered question

Feinsn 2022-06-26

Suppose $T$ is an element of $L({P}_{3}(\mathbb{R}),{P}_{2}(\mathbb{R}))$ is the differentiation map defined by $Tp={p}^{\prime}$. Find a basis of ${P}_{3}(\mathbb{R})$ and a basis of ${P}_{2}(\mathbb{R})$ such that the matrix of T with respect to these basis is

$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

Linear algebraAnswered question

taghdh9 2022-06-26

I have a rectangle with center $({x}_{1},{y}_{1})$ and sides $a,b$ where side $a$ is parallel to axis $Ox$. I want to find a transformation matrix that:

a) converts this rectangle into a square with the same center and side $d$.

b) reflects the rectangle with mirror axis the line $y=sx+c$.

a) converts this rectangle into a square with the same center and side $d$.

b) reflects the rectangle with mirror axis the line $y=sx+c$.

Linear algebraAnswered question

Boilanubjaini8f 2022-06-26

A linear transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{2}$ whose matrix is

$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -3.5+k\end{array}\right)$

is onto if and only if $k\ne $

$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -3.5+k\end{array}\right)$

is onto if and only if $k\ne $

Linear algebraAnswered question

George Bray 2022-06-26

Let $V$ be inner product space.

Let ${e}_{1},...,{e}_{n}$ an orthonormal basis for $V$

Let ${z}_{1},...,{z}_{n}$ an orthonormal basis for $V$

I have to show that the matrix represents the transformation matrix between ${e}_{1},...,{e}_{n}$ to ${z}_{1},...,{z}_{n}$ is unitary.

Let ${e}_{1},...,{e}_{n}$ an orthonormal basis for $V$

Let ${z}_{1},...,{z}_{n}$ an orthonormal basis for $V$

I have to show that the matrix represents the transformation matrix between ${e}_{1},...,{e}_{n}$ to ${z}_{1},...,{z}_{n}$ is unitary.

Linear algebraAnswered question

Izabella Ponce 2022-06-26

If

$A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$

is the matrix representation of a linear transformation

$T:{P}_{2}(x)\to {P}_{2}(x)$

with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+{x}^{2}\}$ then find T. What is the procedure to solve it?

$A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$

is the matrix representation of a linear transformation

$T:{P}_{2}(x)\to {P}_{2}(x)$

with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+{x}^{2}\}$ then find T. What is the procedure to solve it?

Linear algebraAnswered question

Quintin Stafford 2022-06-26

If I am required to compute the full transformation matrix compromising of the following sequence of operations:

rotation by 30 degrees about x-axis

translation by 1, -1, 4 in x, y and z, respectively

rotation by 45 degrees about y axis

Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?

rotation by 30 degrees about x-axis

translation by 1, -1, 4 in x, y and z, respectively

rotation by 45 degrees about y axis

Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?

Linear algebraAnswered question

tasfiaa024 2022-06-25

[1 2 1 \n -1 0 2\n 2 1 -3] reduced the following matrix row echelon form.

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.