# Columns of Matrix Form a Basis?

1. Jan 23, 2012

### puhsyers

Hi everyone,

This not a homework question. I'm reviewing some linear algebra and I found this on a worksheet. I just need a hint on how to approach this problem.

Let β=$\{ v_1,v_2,...,v_n\}$ be a basis for $R^n$. Let M be the matrix whose columns are the basis vectors in β. Do the columns of $M^2$ form a basis?

I've played around with some examples and it seems to be true. I know that M has full rank and hence invertible, therefore, $M^2$ is also invertible and has full rank. But this only means the columns of $M^2$ spans the vector space. How do I show the columns are also linearly independent?

2. Jan 24, 2012

### homeomorphic

If the columns are linearly dependent, then it doesn't have full rank. My definition of rank is the dimension of the space spanned by the columns.