Question: What is Wrong With the Argument?

[lalligagger]

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

Homework Statement

What is wrong with the following argument?

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

Homework Equations

The Fundamental Theorem of Invertible Matrices.

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 C_A and N (A^T) being orthogonal have to do with the equation A^TA\vec{x}=\vec{0}?

 

[HallsofIvy]

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

Homework Statement

What is wrong with the following argument?

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

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

Homework Equations

The Fundamental Theorem of Invertible Matrices.

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 C_A and N (A^T) being orthogonal have to do with the equation A^TA\vec{x}=\vec{0}?

 

[lalligagger]

C_A is the column space of the matrix A and N(A^T) 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 A^TA (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.