Proving result on matrix rank

Question

Is it true that, if $A=QR$ with $Q$ unitary matrix and $R$ an upper triangular matrix, and $A\in\mathbb{C}^{n\times n}$, then the rank of $A$ is the same as that of $R$? And if so, how do I prove it?

Answer 1

$$\text{Rank(A)}=\dim(\text{Im}A)=\dim(Q^{-1}(\text{Im}A))=\dim(\text{Im}Q^{-1}A)=\dim(\text{Im}(R))=\text{Rank}(R).$$ Note that $Q$ is unitary and therefore an isomorphism. You are looking at QR decomposition, or a matrix representation of Gram-Schmidt orthogonalization.