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

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.

• Incorrect.
• Correct! The standard QR factorization will fail if the first $m$ columns of $A$ are dependent (which can happen; for example, the first column of $A$ could be zero).

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

• Correct! The columns of $A^T$ are independent, so the factorization must succeed.
• Incorrect.

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.

• Incorrect.
• Correct! $Q$ will be skinny, so it cannot be an orthogonal matrix.

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