Eigenvalues of a linear transformation (Matrix)

[Iconate]

1. The problem statement, all variables and given/known data
Let T: M22 -> M22 be defined by
T

\[ \left( \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right)\]

=


\[ \left( \begin{array}{cc}
2c & a+c \\
b-2c & d \\
\end{array} \right)\]



Find the eigenvectors of T



3. The attempt at a solution

My main question is, Which matrix am I using to compute my eigenvectors?
Do I need to compute a basis first?

Where this problem differs from my other questions is that I am no longer producing a matrix from my basis vectors which I use to create [T]B

Any insight would be great, thanks.

[Dick]

M22 is a four dimensional vector space, right? A basis is e1=[[1,0],[0,0]], e2=[[0,1],[0,0]], e3=[[0,0],[1,0]] and e4=[[0,0],[0,1]], right? So T must be a 4x4 matrix in that basis, yes? Can you write out what it is in the {e1,e2,e3,e4} basis?

[Iconate]

Yeah I did that, do I find the eigenvectors with each of their matricies?, normally id put my basis vectors INTO a matrix, but I have matricies, i figure they have to go somewhre, just dont know where

[Dick]

I'm not sure I understand that. Just pretend [[a,b],[c,d]] is a 4 vector, [a,b,c,d]. T maps it to another 4 vector [2c,a+c,b-2c,d]. The fact they write these vectors as matrices is just a technicality.

[Iconate]

I figured it out. I have to write T[e1] as a linear combination of the basis vectors.

Ex. T(e1) = [[0,1],[0,0]] = 0*e1 + 1*e2 + 0*e3 + 0*e4
= (0,1,0,0)

And Now i have my vector! Computing this for all ei's will create my matrix P.

[Dick]

I figured it out. I have to write T[e1] as a linear combination of the basis vectors.

Ex. T(e1) = [[0,1],[0,0]] = 0*e1 + 1*e2 + 0*e3 + 0*e4
= (0,1,0,0)

And Now i have my vector! Computing this for all ei's will create my matrix P.

Exactly.