Eigenvalues of a linear transformation (Matrix)

[Iconate]

Homework Statement

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

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 don't 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 e_i'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 e_i's will create my matrix P.

Exactly.