[Math] full QR decomposition- orthonormal basis for the null space of A^{T}

linear algebramatrix decomposition

Show that in the full $QR$ decomposition of the full rank $m\times n$ matrix $A,m ≥ n$, the vectors $q_{n+1}, \ldots, q_m$ are an
orthonormal basis for the null space of $A^T$.

Full $QR$ :$Q\in \mathbb R^{m\times m} , R\in \mathbb R^{m\times n}$

Thanks!

Best Answer

If $A=QR$ is a full decomposition, then $R= \begin{bmatrix} R_1 \\ 0 \end{bmatrix}$ where $R_1$ is invertible.

Then $\ker R^T = \operatorname{sp} \{e_{n+1},\cdots, e_m \}$ and so $\ker A^T = \ker R^T Q^T = Q \ker R^T $ and hence $\ker A^T = \operatorname{sp} \{Qe_{n+1},\cdots, Qe_m \} = \operatorname{sp} \{q_{n+1},\cdots, q_m \}$.

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