22.1 Nullspace, rank and range

1. Determine the nullspace, range and rank of a [latex]m \times n[/latex] matrix of the form

[latex]\begin{align*} A = \begin{pmatrix} S & 0 \\ 0 & 0 \end{pmatrix}. \end{align*}[/latex]

where [latex]S = diag(\sigma_1,...,\sigma_r)[/latex], with [latex]\sigma_1 \geq ... \geq \sigma_r > 0[/latex], and [latex]r \leq \mathrm{min}(m, n)[/latex]. In the above, the zeroes are in fact matrices of zeroes with appropriate sizes.

2. Consider the matrix [latex]uv^T[/latex] with [latex]u \in \mathbb{R}^m, v \in \mathbb{R}^n[/latex].

a. What is the size of [latex]A[/latex]?

b. Determine the nullspace, the range, and the rank of [latex]A[/latex].