For (a) - (d) use the following inner product of $ \mathbb{R}^2 $:\[ \langle x, y \rangle = x^T A y \]where $ A = \begin{bmatrix} 3 & -2 \\ -2 & 5 \end{bmatrix} $. Let $ x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} $ and $ y = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $.(b) Using $ \langle \cdot, \cdot \rangle $ defined above compute the following: i) $ \| x \| $ ii) $ \| y \| $

Given the matrix and vector,, , and .The inner product is defined as .The aim is to find the values…

Answered: For (a). (d) use the following inner…

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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Question

For (a) - (d) use the following inner product of $ \mathbb{R}^2 $:

\[ \langle x, y \rangle = x^T A y \]

where $ A = \begin{bmatrix} 3 & -2 \\ -2 & 5 \end{bmatrix} $. Let $ x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} $ and $ y = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $.

(b) Using $ \langle \cdot, \cdot \rangle $ defined above compute the following:
i) $ \| x \| $

ii) $ \| y \| $

Expert Solution

Step 1: Given the information

Given the matrix and vector,

$A equals open square brackets table row 3 cell negative 2 end cell row cell negative 2 end cell 5 end table close square brackets$ , $x equals open square brackets table row 1 row 1 end table close square brackets$ , and $y equals open square brackets table row 0 row 1 end table close square brackets$ .

The inner product is defined as $open angle brackets x comma space y close angle brackets equals x to the power of T A y$ .

The aim is to find the values of $open double vertical bar x close double vertical bar$ and $open double vertical bar y close double vertical bar$ .