Recent questions in Matrix transformations

Matrix transformations
Answered

Nylah Hendrix
2022-07-02

Now my questions:

- Ok so this vector is invariant. So what? (in my case for attitude determination algorithm I even less understand what this could give me as useful information)

- how does a simple $4\times 4$ matrix actually represent a transformation?

Matrix transformations
Answered

uri2e4g
2022-07-02

Matrix transformations
Answered

kramberol
2022-07-02

$A=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -24& -29& -18\end{array}\right)$

with its eigenvalues of: $-16.3$, $-0.844+0.871j$ and $-0.844-0.871j$

Matrix A can be diagonalized to the classical known form of

${A}_{\text{diag}}=\left(\begin{array}{ccc}-0.844+0.871j& 0& 0\\ 0& -0.844-0.871j& 0\\ 0& 0& -16.3\end{array}\right)$

with a vandermone transformation matrix

The Question is I need a transformation matrix to transform matrix A to the form of matrix Ad (with no complex elements)

${A}_{d}=\left(\begin{array}{ccc}-0.844& 0.871& 0\\ -0.871& -0.844& 0\\ 0& 0& -16.3\end{array}\right)$

and this form called a quasi-diagonal matrix

Matrix transformations
Answered

Ciara Mcdaniel
2022-07-01

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

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

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

Matrix transformations
Answered

hornejada1c
2022-07-01

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

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

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 ${A}^{-1}$?

Matrix transformations
Answered

Cristopher Knox
2022-07-01

Matrix transformations
Answered

Kristen Stokes
2022-07-01

(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 $T(-{x}^{2}-3x+2)$

Matrix transformations
Answered

spockmonkey40
2022-06-30

Matrix transformations
Answered

kokoszzm
2022-06-30

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

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

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?

Matrix transformations
Answered

Jameson Lucero
2022-06-29

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

[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

Matrix transformations
Answered

veneciasp
2022-06-29

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

How to find the standard matrix?

Matrix transformations
Answered

bandikizaui
2022-06-29

Matrix transformations
Answered

Devin Anderson
2022-06-29

Matrix transformations
Answered

Makayla Boyd
2022-06-28

(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?

Matrix transformations
Answered

Dania Mueller
2022-06-27

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

Matrix transformations
Answered

Poftethef9t
2022-06-27

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

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?

Matrix transformations
Answered

Feinsn
2022-06-26

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

Matrix transformations
Answered

taghdh9
2022-06-26

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

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

Matrix transformations
Answered

Boilanubjaini8f
2022-06-26

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

is onto if and only if $k\ne $

Matrix transformations
Answered

George Bray
2022-06-26

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.

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.