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.

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.

part 2 is just rephrasing part 1 of your question in terms of matrices. for part 1, since T is not an isomorphism, $\displaystyle \ker T \neq (0).$ thus $\displaystyle \dim \ker T \geq 1.$ choose a basis for $\displaystyle \ker T,$ say $\displaystyle B_1=\{v_1, \cdots , v_k \},$
and extend it to a basis $\displaystyle B$ for $\displaystyle V.$ then $\displaystyle T(v_1)=\bold{0}$ is the first column of $\displaystyle [T]_B. \ \Box$