I'm trying to understand the proof of the following statement:
If $A$ has independent columns then $A^TA$ is invertible.
Please keep in mind that these structured are based on real field.
The proof starts with following equation:
Then it is stated, that for $A^TA$ to be invertible, $x$ must be the zero vector.
In other words, for $A^TA$ to be invertible, it's null space must be the zero vector.
I found this answer, but I couldn't completely understand it. The author of the answer defined variables: $A$ being a matrix containing columns of $a_n$, $x$ being a vector containing columns of $(x_n)^T$, and $Ax$ being equal to $0$.
Then it's obvious that $x_na_n=0$, but author states that unless $x_n=0$, the columns will not be linearly independent.
I couldn't understand this. If $A$ has linearly independent columns, then how will it change at all when being multiplied by $x$? Maybe they mean $Ax$, but still even if $x$ is zero vector, isn't $Ax$ just a simple linear combination of $A$ and $x$? (And if that is true then it's definitely not linearly independent).