Prove AtA is Nonsingular: Invertible Matrix Homework

[roam]

Homework Statement

If A is a an mxn matrix and its column vectors are linearly independent.

Prove that the matrix A^tA is nonsingular. Hint: Use the fact that it is sufficient to show that null(A^tA) = {0}

Homework Equations

The Attempt at a Solution

I'm new to this topic & I don't understand the hint given and how exactly to use it to prove the question...

I know that in order for a matrix to be nonsingular/invertible it has to be squre (m=n) and when you multiply the a matrix by its transpose, the resulting matrix would be square.
I'm also thinking about the properties of fundamental spaces of matrices that: row(A) = null(A) and col(A)=null(A^t) (therefore null(A^tA) = row(A).col(A)?)

Any help would be much appreciated :)

Cheers.

 

[Dick]

If the columns of A are linearly independent, doesn't that mean null(A)={0}? Now suppose A^(T)A were singular and think about x^(T)A^(T)*Ax, where x is a column vector in R^n.