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
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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:
\[ U^T U = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix} \]
\[ UU^T = \begin{bmatrix} \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \end{bmatrix} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F018ccba1-bef4-4bce-a1d3-c440c71f992e%2F9380057f-e086-49e0-800a-77977e69d9a6%2F7ztc0lf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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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