Prove Au and Av Linearly Independent

[cookiesyum]

Homework Statement

Let u_1...u_n be linearly independent column vectors in R^n and A an invertible n x n matrix. Prove that the vectors Au_1...Au_n are linearly independent.

Homework Equations

The Attempt at a Solution

It is easy to prove this using scalars and the definition of linear independence. But, then why is this relevant to invertible matrices? Is there a way to prove this using column spaces, row spaces, null spaces, etc?

 

[quasar987]

It is not true if A is not invertible!