gibson:teaching:fall-2016:iam961:hw1
====== Differences ====== This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
gibson:teaching:fall-2016:iam961:hw1 [2016/09/02 08:25] gibson |
gibson:teaching:fall-2016:iam961:hw1 [2016/09/07 08:56] (current) gibson |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== IAM 961 HW1 ====== | ====== IAM 961 HW1 ====== | ||
| - | **1** Prove that any linear map $\mathcal{L} : \mathbb{C}^n \rightarrow \mathbb{C}^m$ can | + | **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 $\|A B \|_p \leq \|A\|_p \|B\|_p$. | + | **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 | + | **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 | //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 | has the form $A^{-1} = I + \alpha u v^*$ for some scalar $\alpha$, and give an expression | ||
gibson/teaching/fall-2016/iam961/hw1.1472829903.txt.gz · Last modified: 2016/09/02 08:25 by gibson
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
