$Q$ is an orthogonal matrix. $R$ is an upper triangular matrix. $A \in \mathbb{R}^{m\times n}$ with $m > n$ and its QR-Factorizations is $A = QR$. Show that if $A$ has full rank, then the diagonal elements of $R$ are non-zero. Show also that the first $n$ columns of $Q$ are an orthonormal basis of the column space of $A$.
I tried to prove that, but with no success. Can someone help me?
Best Answer
Hint: if $R$ is upper triangular and $R_{kk} = 0$, then its first $k$ columns must be linearly dependent, which makes $R$ have rank $< n$.
By "the spanning of $A$" you mean "the column space of $A$", i.e. the span of the columns of $A$.