IAM 961 HW1

1. Prove that any linear map $\mathcal{L} : \mathbb{C}^n \rightarrow \mathbb{C}^m$ can written as an $m \times n$ matrix. (Hint: let $y = \mathcal{L}(x)$ . Express $x$ as a linear combination of the canonical basis vectors $\{e_j\}$ . Substitute that into $y = \mathcal{L}(x)$ , then use linearity to rewrite the right-hand-side of this equation as a linear combination of vectors $\mathcal{L}(e_j)$ . Now take the inner product of both sides of this equation with $e_i$ . That should give you $y_i = \sum_{j=1}^n L_{ij} x_j$ for some matrix $L$ .)

2. Prove that $\|A B \|_p \leq \|A\|_p \|B\|_p$ .

3. If $u$ and $v$ are $m$ -vectors the matrix $A = I + uv^*$ is known as a rank-one perturbation of the indentity. Show that if $A$ is nonsingular, then its inverse has the form $A^{-1} = I + \alpha u v^*$ for some scalar $\alpha$ , and give an expression for $\alpha$ . For what $u$ and $v$ is $A$ singular? If it is singular, what is $\text{null}(A)$ ? (Trefethen exercise 2.6).

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IAM 961 HW1

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