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[M] Generate random vectors $\mathbf{x}, \mathbf{y},$ and $\mathbf{v}$ in $\mathbb{R}^{4}$ with integer entries (and $\mathbf{v} \neq \mathbf{0} ),$ and compute the quantities$\left(\frac{\mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v},\left(\frac{\mathbf{y} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}, \frac{(\mathbf{x}+\mathbf{y}) \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}, \frac{(10 \mathbf{x}) \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$Repeat the computations with new random vectors $\mathbf{x}$ and $\mathbf{y} .$ What do you conjecture about the mapping $\mathbf{x} \mapsto T(\mathbf{x})=$ $\left(\frac{\mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}(\text { for } \mathbf{v} \neq \mathbf{0}) ?$ Verify your conjecture algebraically.

Generate the following random vectors in $\mathbb{R}^{4}$ with integer entries.$>>\mathbf{X}=\operatorname{randi}([-5,5], 4,1)$$\mathrm{x}=$3$-1$1$-5$$>>\mathbf{y}=\operatorname{randi}[[-5,5], 4,1)$$y=$035$>>v=\operatorname{randi}[[-5,5], 4,1)$$v=$$-4$10$-5$Compute the quantities $\left(\frac{\mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v},\left(\frac{\mathbf{y} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}, \frac{(\mathbf{x}+\mathbf{y} \cdot \mathbf{v})}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}, \frac{10 \mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$ using the below$>>\mathrm{p}=(\operatorname{dot}(x, V) / \operatorname{dot}(\mathrm{v}, \mathrm{v}))^{*} \mathrm{v}$ans $=$$-1.1429$0.28570$-1.4286$$>>\mathrm{q}=(\operatorname{dot}(y, \mathrm{v}) / \operatorname{dot}(\mathrm{v}, \mathrm{v}))^{*} \mathrm{v}$ans $=$0.4762$-0.1190$00.5952$>>r=(\operatorname{dot}(X+y, V) / \operatorname{dot}(v, v))^{*} V$ans $=$$-0.6667$0.16670$-0.8333$$>>\mathrm{S}=\left(\operatorname{dot}\left(10^{*} \mathrm{x}, \mathrm{V}\right) / \operatorname{dot}(\mathrm{v}, \mathrm{v})\right)^{*} \mathrm{V}$ans $=$$-11.4286$2.85710$-14.2857$Note the following:$>>p+q$ans $=$$-0.6667$0.16670$-0.8333$This is $\mathrm{r}$ .Also$>>10^{*} \mathrm{p}$ans $=$- 11.42862.8571-14.2857This is s.Here it is demonstrated that mapping $\mathbf{x} \mapsto T(\mathbf{x})=\left(\frac{\mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}(\text { for } \mathbf{v} \neq 0)$ is a linearTransformation as $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$ and $T(c \mathbf{x}+\mathbf{y})=c T(\mathbf{x}),$ with $c=10$The algebraic verification is as follows:Let $x$ and $y$ be in $\mathbb{R}^{n}$ , and let $c$ be any scalar.Then$\begin{aligned} T(\mathbf{x}+\mathbf{y}) &=\left(\frac{(\mathbf{x}+\mathbf{y}) \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v} \\ &=\left(\frac{(\mathbf{x} \cdot v)+(\mathbf{y} \cdot v)}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v} \\ &=\left(\frac{\mathbf{x} \cdot v}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}+\left(\frac{\mathbf{y} \cdot v}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v} \\ &=T(\mathbf{x})+T(\mathbf{y}) \end{aligned}$and$\begin{aligned} T(c \mathbf{x}) &=\left(\frac{(c \mathbf{x}) \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v} \\ &=\left(\frac{c(\mathbf{x} \cdot v)}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v} \\ &=c\left(\frac{\mathbf{x} \cdot v}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v} \\ &=c T(\mathbf{x}) \end{aligned}$Thus, $x \mapsto T(x)$ is a linear transformation.

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 1

Inner Product, Length, and Orthogonality

Vectors

Johns Hopkins University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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In mathematics, a vector (…

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[M] Construct a random $4 …

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Prove the following identi…

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Suppose $\mathbf{u}$ and $…

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Cross product equations Su…

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Okay. So in this example, we want to see how these four transformations will affect the determines off A when we use random former for matrices. So I created a love another program here, where we will generate random matrices depending on this shape. Yeah, so if you want to use it on five by five and 656 matrix Just changed history. Five comma five and six. Common six. Then what this will do is that this will find the ratio between eight transfers on A. So it's going to wipe transfers and eight to the ratio between negative, eh? And eh, The ratio between two. A two and ratio of 10 8 to a and then everything worry about what's called us. But that's what it does. So what you find out is that the determined a transpose should be equal to two determinant off, eh? Determined off negative. A is equal to negative one times by and determines all, eh? So here I'm just gonna make a random a general n by N matrix. And then this one will be true to the end determinants. Oh, hey, this is 10 to the end tournaments away and in this case because this is for for so many years, for this will just be one to the end. So it's gonna be equal to determine off A is going to be too to the form, so you'll get 16 and this is tentative. Fall and you'll you'll see this when you perform this program.

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