Can Linear Independence Affect Pivot Positions in Matrix Columns?

[Cali210]

Homework Statement

So the question is, Prove the following:

Let A be an n x n matrix. If there exists a vector v in Rn that is not a linear
combination of the columns of A, then at least one column of A is not a pivot column.

Homework Equations

The only relevant theorem I think is the invertible matrix theorem, which i attached.
I also attached theorem 4 (book has different names)

The Attempt at a Solution

So far, I started with

- Let A be a n x n matrix and v be a vector in Rn that is not a linear combination of the columns of A
- then there is not a pivot position in every row of rref A (theorem 8 not g to not c)
- then there is at most n-1 pivot positions (out of n rows)
- then at least one column of rref A is not a pivot position (square matrix).

the question i have is, am i allowed to jump from my first sentence to my second sentence without any justification?

For some reason, I am thinking that I'm supposed to go from (theorem 4 not b to not a)
and then, since theorem 4a is equivalent to theorem 8g, then jump from (theorem 8 not g to not c). But am i allowed to use and jump from theorem 4 to theorem 8? since theorem 4 is for a m x n matrix, while theorem 8 is n x n matrix?

Also, in theorem 4 there's a statement " each b in Rm is a linear combination of the columns of A", which is the assumption i started with. What would be the equivalent statement in theorem 8, if there is any?

Sorry, this is probably a basic proof question, but I'm just horrible at proving something, so I wanted to make sure I was on the right path.

 

[phiiota]

What about span?