Linear algebra help: Linear independence

1. Feb 22, 2012

epkid08

1. The problem statement, all variables and given/known data
Let A be an m x n matrix of rank n. Suppose $$v_1, v_2, ..., v_k \in \mathbb{R}^n$$ and $$\{v_1, v_2, ..., v_k\}$$ is linearly independent. Prove that $$\{Av_1, Av_2, ..., Av_k\}$$ is likewise linearly independent.

2. Relevant equations

3. The attempt at a solution
It says I need to use rank(A) = n, but I'm not sure how to use that info.

2. Feb 22, 2012

Dick

Rank n tells you that if you have a basis $e_1, e_2, ..., e_n$ of $\mathbb{R}^n$ then $A(e_1), A(e_2), ..., A(e_n)$ are linearly independent. I.e. A takes R^n into an n-dimensional subspace of R^m.

3. Feb 23, 2012

Deveno

have you proven a subset of a linearly independent set is also linearly independent?

if so, extend the v's to a basis.