Linear Transformation Question: Solving for Im(T) in R^4 Dimension Space

[stud]

I need help with question from homework in linear algebra.

This question (linear transformation):
http://i43.tinypic.com/15reiic.gif

According to theorem dimensions:
dim(V) = dim(Ker(T)) + dim(Im(T)).

dim(Ker(T))=2.
dim(V) in R^4, meaning =4.

We can therefore conclude that dim(Im(T))=2.
But how can find this specific linear transformation?

Please help me to solve this problem.
Thanks.

 

[lanedance]

So let's write the transform as a 4x4 matrix A, giving 16 unknowns.

Now you you know Ax=0 for, x in the kernal. This gives you 8 equations, and is equivalent to mappint the plane contain the 2 kernal vectors to zero.

Clearly you still have more unknowns, so you will need to make some assumptions.

One of the simpler ones, might be to try and find the "projection matrix" which maps components in the kernal plane to zero, but leaves components perpindicular to that plane unchanged