I am confused about the bi-conditional statement I outlined in the title.
I'll write it again here: (This was part of a solution to an assignment question)
$A$ has full column rank if and only if the symmetric matrix $B=A^TA$ is positive definite.
The definition of column rank that I am aware of states that a $m \times n$ Matrix $A$ has full column rank if each of the columns are linearly independent. So it would be full rank if $rank(A) = n$ in this case.
And I guess, the dimension of $B = A^TA$ is $n \times n$. So it would be a square matrix that is full column rank. Now going back to the statement, why can we say that this matrix $B$ is symmetric, let alone a positive definite?