Proofs using column vectors and invertible matrix

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!

Re: Proofs using column vectors and invertible matrix

Quote:

Originally Posted by

**widenerl194** 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.

Okay, what **is** the "Invertible Matrix Theorem"?

Quote:

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}.

This is confusing, you said above that v_{1}, v_{2}, ...., v_{n} were the columns of A. Now you are saying they are the standard basis for R^{n}.

Which do you want?

Quote:

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!

Then you say "Let c be in the reals" but talk about c_{1}, c_{2}, ..., c_{n}. Did you mean "c_{i}" rather than just i?

And what do you mean by "[c_{1}] = the zero vector = [0]"? What does the "[ ]" indicate?

Re: Proofs using column vectors and invertible matrix

The invertible matrix theorem says:

Let A be an n x n matrix, I the n x n identity matrix, and θ be the zero column vector in R. The following are equivalent:

1. A is invertible

2. The reduced row echelon form of A is I

3. For any b in R^n, the equation Ax=b has exactly one solution

4. The equation Ax=θ has only x=θ as a solution

The v1 v2 ... vn should be the columns