# [Math] Why must the basis vectors be orthogonal when finding the projection matrix.

linear algebralinear-transformationsorthogonality

I was reading my coursebook on linear algebra, and i noted that one of the examples in the book mentioned that it was necessary for the basis of a subspace $V$ to be orthogonal, in order to determine the projection matrix for $V$.

My question is then; why is it necessary to figure out if the vectors that span the subspace are orthogonal in order to determine the projection matrix onto the subspace?

Is it because, in order to determine the projection matrix you have to turn the basis into a orthonormal basis, and thus it is useful to check if they are orthogonal, and that way you could avoid converting from a normal basis to an orthogonal basis?

On the other hand, if you assemble the basis vectors as columns of the matrix $A$, the projection onto their span is given by $A(A^TA)^{-1}A^T$, which only requires that $A$ have full rank, i.e., that the vectors are linearly independent, which is true since you’re working with a basis of the subspace.