= Solution
A <QR decomposition> of an $m\times n$ matrix, $m\ge n$, is $A=QR$, where $Q$ is an $m\times m$ <orthogonal matrix> and $R$ is $m\times n$ and upper trapezoidal: $R_{ij}=0$ for $i>j$. In particular its bottom $m-n$ rows vanish.
Successive left <Givens rotations> annihilate the entries below the diagonal, column by column. When working on column $j$, rotate rows $j$ and $i>j$; their earlier columns are already zero, so the earlier eliminations persist. If $G_k\cdots G_1A=R$, then
$$
\boxed{Q=G_1^T\cdots G_k^T,\qquad A=QR.}
$$
A product of <orthogonal matrices> is orthogonal. The construction remains valid for a rank-deficient matrix, although uniqueness is not asserted. This is <QR decomposition by Givens rotations>.
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