Linear transformation, Tx=Ax

mattie

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Hi guys!

I am struggeling with this task and would appreciate any help at all!

Let T be a linear imaging T : R2 → R2s, which is so that:

. . .\(\displaystyle T\binom{2}{1}\, =\, \binom{-2}{1}\). . .og. . .\(\displaystyle T\binom{1}{1}=\binom{1}{1}\)

a) Find a 2x2 matrix A thus Tx=Ax for all x ∈ R^2
b) Find two eigenvalues and two eigenvectors for A

c) Calculate:

. . .\(\displaystyle (A^5\, +\, A^3\, +\, A)\binom{2}{1}\)

Would love any kind of help with this!
 

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Hi guys!

I am struggeling with this task and would appreciate any help at all!

Let T be a linear imaging T : R2 → R2s, which is so that:

View attachment 7726
a) Find a 2x2 matrix A thus Tx=Ax for all x ∈ R^2
b) Find two eigenvalues and two eigenvectors for A
c) Calculate: View attachment 7727

Would love any kind of help with this!
Hi. This is a tutoring web site, so we like to see what you've tried or thought about so far.

Here is a link to the forum guidelines. :cool:
 
One obvious way is to let the 2 by 2 matrix be \(\displaystyle \begin{pmatrix}a & b \\ c & d \end{pmatrix}\). Then saying that \(\displaystyle T\begin{pmatrix}2\\ 1\end{pmatrix}= \begin{pmatrix}-2 \\ 1\end{pmatrix}\) means that \(\displaystyle \begin{pmatrix}a& b \\ c & d \end{pmatrix}\begin{pmatrix}2 \\ 1\end{pmatrix}= \begin{pmatrix}2a+ b \\ 2c+ d \end{pmatrix}= \begin{pmatrix}-2 \\ 1\end{pmatrix}\) so that we the two equations 2a+ b= -2 and 2c+ d= 1. Do the same thing with the other equation to get four linear equations to solve for a, b, c, and d.
 
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