Linear Algebra Done Right · Problem 3C.7

The key is to build a basis of $V$ that respects the structure of $T$'s image. Start by deciding what you want the first column of the matrix to look like—either all zeros (if $T$ is zero) or a 1 in position $(1,1)$—then construct $v_1$ accordingly. For the remaining basis vectors $v_2, \ldots, v_m$, choose them to span the kernel of $T$ or to be orthogonal to $w_1$ in the appropriate sense, ensuring their images don't interfere with the first row. This is a dimension-counting argument: you have freedom in choosing $V$'s basis, so use it strategically to zero out unwanted entries.

The key is to build a basis of $V$ that respects the structure of $T$'s image. Start by deciding what you want the first column of the matrix to look like—either all zeros (if $T$ is zero) or a 1 in position $(1,1)$—then construct $v_1$ accordingly. For the remaining basis vectors $v_2, \ldots, v_m$, choose them to span the kernel of $T$ or to be orthogonal to $w_1$ in the appropriate sense, ensuring their images don't interfere with the first row. This is a dimension-counting argument: you have freedom in choosing $V$'s basis, so use it strategically to zero out unwanted entries.