True or False?

Let $A$ be any $n\times n$ invertible matrix. So, without any loss of generalization, we can suppose that $A$ is the matrix representation of any bijective Linear Transformation $T: X \to Y$ , where the dimension of both, $X$ and $Y$ is $n$ , which are both Vector Spaces. As vector space it is, there exist $n$ linearly independent vectors in $Y$ which are its Basis. By Gram-Schmidt's process, there is an orthonormal set of vectors which are basis of $Y$ and other set of vectors for $X$ . And then is clear that, because of the bijectivity of $T$ there are $n$ vectors ${v_{1}, v_{2},... v_{n}} \in X$ such that $Av_{1} \perp Av_{2} \perp... \perp Av_{n}$