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22 EXERCISES

22.1 Nullspace, rank and range

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

    \begin{align*} A = \begin{pmatrix} S & 0 \\ 0 & 0 \end{pmatrix}. \end{align*}

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

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

a. What is the size of A?

b. Determine the nullspace, the range, and the rank of A.

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