# FTIM question

1. Mar 25, 2009

### lalligagger

Sorry all my vectors look like superscripts, don't know what that's about.

1. The problem statement, all variables and given/known data

What is wrong with the following argument?

Let A be an arbitrary m x n matrix. The vector A$$\vec{x}$$ is obviously in CA so it can't be in N(AT) unless it's the zero vector, since CA is orthogonal to N(AT). Thus the only solution to ATA$$\vec{x}$$=$$\vec{0}$$ is $$\vec{x}$$=$$\vec{0}$$ and ATA is an invertible matrix (by FTIM).

2. Relevant equations

The Fundamental Theorem of Invertible Matrices.

3. The attempt at a solution

I don't know if I understand the (incorrect) reasoning behind this argument. Mainly, I don't understand the connection between the end of the second sentence and the first half of the last sentence. What does CA and N (AT) being orthogonal have to do with the equation ATA$$\vec{x}$$=$$\vec{0}$$?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 25, 2009

### HallsofIvy

It would help a lot if you would tell us (1) what CA and N(AT) mean, and (2) what this argument is supposed to prove.

3. Mar 25, 2009

### lalligagger

CA is the column space of the matrix A and N(AT) is the null space of A transpose. Sorry, I thought that was standard notation.
The argument says that given an arbitrary matrix A, the matrix ATA (the matrix you get when you multiply A by A transpose on the left) is invertible. The point of the problem is to recognize that this isn't true and find the faulty reasoning.