b. Compute PROJwY and (UU^(t))y **Vector Spaces and Orthogonality**Consider the following vectors and matrices:Given vectors:\[ \mathbf{y} = \begin{bmatrix} 3 \\ 5 \\ 1 \end{bmatrix}, \]\[ \mathbf{u}_1 = \begin{bmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{bmatrix}, \]\[ \mathbf{u}_2 = \begin{bmatrix} \frac{2}{3} \\ \frac{1}{3} \\ -\frac{2}{3} \end{bmatrix}, \]and the vector space $ W $ defined as:\[ W = \text{Span} \{\mathbf{u}_1, \mathbf{u}_2\} \]To complete parts (a) and (b):(a) Define matrix $ U $ as:\[ U = \begin{bmatrix} \mathbf{u}_1 & \mathbf{u}_2 \end{bmatrix} = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & -\frac{2}{3} \end{bmatrix} \]We will compute the following:\[ U^T U \text{ and } UU^T \]Evaluate the expressions and simplify your answers to find:\[ U^T U = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix} \]\[ UU^T = \begin{bmatrix} \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \end{bmatrix} \]

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Description: b. Compute PROJwY and (UU^(t))y **Vector Spaces and Orthogonality**Consider the following vectors and matrices:Given vectors:\[ \mathbf{y} = \begin{bmatrix} 3 \\ 5 \\ 1 \end{bmatrix}, \]\[ \mathbf{u}_1 = \begin{bmatrix} \frac{1}{3} \\ \frac{2}{3} \\

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Answered: 3 Let y = 5 U2 = and W = Span (u1 u2}.…

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3 Let y = 5 U2 = and W = Span (u1 u2}. Complete parts (a) and (b). 1. 3 a. Let U = u, u, Compute U'U and UU. U'U-and UU =(Simplify your answers.) 2/31/3 2/3 1/3

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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b. Compute PROJwY and (UU^(t))y

**Vector Spaces and Orthogonality**

Consider the following vectors and matrices:

Given vectors:

\[ \mathbf{y} = \begin{bmatrix} 3 \\ 5 \\ 1 \end{bmatrix}, \]

\[ \mathbf{u}_1 = \begin{bmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{bmatrix}, \]

\[ \mathbf{u}_2 = \begin{bmatrix} \frac{2}{3} \\ \frac{1}{3} \\ -\frac{2}{3} \end{bmatrix}, \]

and the vector space $ W $ defined as:

\[ W = \text{Span} \{\mathbf{u}_1, \mathbf{u}_2\} \]

To complete parts (a) and (b):

(a) Define matrix $ U $ as:

\[ U = \begin{bmatrix} \mathbf{u}_1 & \mathbf{u}_2 \end{bmatrix} = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & -\frac{2}{3} \end{bmatrix} \]

We will compute the following:

\[ U^T U \text{ and } UU^T \]

Evaluate the expressions and simplify your answers to find:

\[ U^T U = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix} \]

\[ UU^T = \begin{bmatrix} \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \end{bmatrix} \]

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