Proving the Spanning Property of Linearly Independent Columns in Lin Alg

[tandoorichicken]

Problem: Explain why the columns of $A^2$ span $\mathbb{R}^n$ whenever the colums of A are linearly independent.

By the theorem given in that section of the text, it is a logically equivalent fact that if the columns of $A^2$ are linearly independent, then they span $\mathbb{R}^2$ or
$$\mathbb{R}^2=Span( \vec{a}_1 , \vec{a}_2 )$$.

How do I expand this definition from $\mathbb{R}^2$ to $\mathbb{R}^n$?

 

[AKG]

If B is an nxn real matrix, then what can you say about whether or not its columns span Rⁿ if its columns are linearly independent. Don't worry about B being a matrix. You know that it since it is nxn, it gives you n linearly independent columns, so you should know something about whether those columns span Rⁿ. Once you know this, you should be able to say something about conditions on x if Bx = 0. You should also be able to say something about the invertibility of B. Can you get this far?