$\newcommand{\ones}{\mathbf 1}$

A skinny matrix whose columns form an orthonormal set is called


Suppose $A \in {\mathbf R}^{m \times n}$ is fat (i.e., $m < n$) and full rank.

The standard QR factorization of $A$ cannot fail.

The standard QR factorization of $A^T$ cannot fail.

If the standard QR factorization of $A$ succeeds, the resulting $Q$ is orthogonal.

If the standard QR factorization of $A^T$ succeeds, the resulting $Q$ is orthogonal.


Suppose the columns of $A=[a_1 ~ \cdots ~a_k]$ are independent, with QR factorization $A=[q_1 ~ \cdots ~q_k]R$, and the fourth column of $R$ has only one nonzero entry. This implies


Suppose $A \in {\mathbf R}^{m \times n}$.

The statement $\mathcal R(A^T) + \mathcal N(A) = {\mathbf R}^n$

The statement $\mathcal R(A) \perp \mathcal N(A)$