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--- gibson:teaching:fall-2016:iam961:hw1 [2016/09/02 08:25]
gibson
+++ gibson:teaching:fall-2016:iam961:hw1 [2016/09/07 08:56] (current)
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@@ Line 2: / Line 2: @@
 **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 $L$.)
+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 $\|A B \|_p \leq \|A\|_p \|B\|_p$.

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