Here it is:

Let A be an n x n matrix. Denote it's columns by v_{1}, v_{2}, ... vn. Keep in mind that these are column vectors in R^{n}.

a. Prove that A is invertible if and only if {v_{1}, v_{2}, ... v_{n}} is linearly independent. We are supposed to use the Invertible Matrix Theorem here.

This is what I have so far:

Let A be invertible. We want to show {v_{1}, v_{2}, ... v_{n}} is linearly independent. {v_{1}, v_{2}, ... v_{n}} is the standard basis for R^{n}. Let c be in the reals. Then set c_{1}v_{1}+ c_{2}v_{2}+ ... + c_{n}v_{n}= the zero vector. Then

[c_{1}] = the zero vector = [0]

[c_{2}] [0]

[c_{n}] [0]

Thus, {v_{1}, v_{2}, ... v_{n}} is linearly independent.

I'm not sure if this is right or how to complete the second half of the proof. Any help would be appreciated!