Verify that the mapping T : R4 -> R2 defined by
T (x1; x2; x3; x4) = (x3 - x2; 5x1 + 3x4)
is a linear transformation.
Find a basis for Im (T ) and Ker (T ).
Showing linearity:
Choose 2 vectors in .
Choose
We want to show that :
Can you do that? It's just writing out.
Observe that ker . Hence v satisfies .
Find two independant vectors in that satisfy these relations and you have a basis for ker(T).
Find 2 more independant vectors: and you have a basis for Im(T)
(Dim(Im(T))+ Dim(ker(T)) = 4).