Linear Algebra - How to represent this transformation?

[zeion]

Homework Statement

Given that T is the linear operator on R3 with T(1,1,1) = (0,0,1), T(1,1,0) = (1,2,1), T(1,0,0) = (0,-1,0), determine the eigenvalues of T and a corresponding eigenvector for each eigenvalue.

Homework Equations

The Attempt at a Solution

I know how to find eigenvalues and vectors but usually the matrix is given.. I'm not sure how to represent these transformations as one whole matrix.. or do I need to do them separately?

 

[vela]

You can get the columns of the matrix by applying T to the basis vectors, so you just need to figure out how to calculate T(1,0,0), T(0,1,0), and T(0,0,1) to find the matrix.

 

[zeion]

Okay in that case I can make an augmented matrix with the given relations and row reduce the left side to be the standard basis and the augmented side will be affect of T on the standard basis? Then I can transpose that to get the matrix?

 

[vela]

If you can find a, b, and c such that

(0,1,0) = a(1,1,1)+b(1,1,0)+c(1,0,0)

then

T(0,1,0)=T[a(1,1,1)+b(1,1,0)+c(1,0,0)]=...

Use T's linearity to evaluate the RHS. Do the same for the vector (0,0,1).

 

[zeion]

I don't really understand.. How can I know what T does just by looking at the vectors that T is applied to..?

 

[vela]

Like I said, use the fact that T is linear to evaluate the RHS.

 

[zeion]

Okay so for
(0,1,0) a = 0, b= 1, c=-1
(0,0,1) a = 1, b = -1, c = 0