# Thread: linear transformations

1. ## linear transformations

I need help with this problem:

1. let V be vector space with a finite dimension and let T:V to V be linear transformation. prove that if T is not isomorphism, so there is a basis B of V that [T]B is a matrix with a zero column.

2. Prove that if A is singular so it's similar to a matrix with a zero column.

Thanks ahead

2. Originally Posted by omert
I need help with this problem:

1. let V be vector space with a finite dimension and let T:V to V be linear transformation. prove that if T is not isomorphism, so there is a basis B of V that [T]B is a matrix with a zero column.

2. Prove that if A is singular so it's similar to a matrix with a zero column.

Thanks ahead
part 2 is just rephrasing part 1 of your question in terms of matrices. for part 1, since T is not an isomorphism, $\ker T \neq (0).$ thus $\dim \ker T \geq 1.$ choose a basis for $\ker T,$ say $B_1=\{v_1, \cdots , v_k \},$

and extend it to a basis $B$ for $V.$ then $T(v_1)=\bold{0}$ is the first column of $[T]_B. \ \Box$