6. Find the following. a. Find the B-coordinate vector of s(t) = 4+5 sin?(t) given B = {1, sin(t), sin²(t), sin (t)}.
6. Find the following. a. Find the B-coordinate vector of s(t) = 4+5 sin?(t) given B = {1, sin(t), sin²(t), sin (t)}.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Question
Find the B- coordinate vector of s(t)
![**6. Find the following:**
a. Find the \( B \)-coordinate vector of \( s(t) = 4 + 5 \sin^2(t) \) given \( B = \{1, \sin(t), \sin^2(t)\} \).
\[ s(t) = 4 + 5 \sin^2(t) \]
\[ B = \{1, \sin(t), \sin^2(t)\} \]
b. Find \( x \), given the basis \( B = \left\{ \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} -4 \\ -7 \\ 0 \end{bmatrix} \right\} \) and \([x]_B = \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}\).
\[ \vec{x} = 2 \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} + (-1) \begin{bmatrix} 3 \\ 2 \\ -2 \end{bmatrix} + 0 \begin{bmatrix} -4 \\ -7 \\ 0 \end{bmatrix} \]
\[ = \begin{bmatrix} (1)(2) + (3)(-1) + 4(0) \\ -2(2) + 2(-1) - 7(0) \\ 1(2) - 2(1) + 0(0) \end{bmatrix} = \begin{bmatrix} -1 \\ -6 \\ 4 \end{bmatrix} \]
This transcription provides the mathematical procedures for finding the \( B \)-coordinate vector and the vector \( x \) in relation to the given bases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe358470e-7791-4930-99ad-4bf2c4c86737%2F54e50a67-0b51-440e-ad4e-5f7a73954f37%2Fzs0j7fg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**6. Find the following:**
a. Find the \( B \)-coordinate vector of \( s(t) = 4 + 5 \sin^2(t) \) given \( B = \{1, \sin(t), \sin^2(t)\} \).
\[ s(t) = 4 + 5 \sin^2(t) \]
\[ B = \{1, \sin(t), \sin^2(t)\} \]
b. Find \( x \), given the basis \( B = \left\{ \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ -2 \end{bmatrix}, \begin{bmatrix} -4 \\ -7 \\ 0 \end{bmatrix} \right\} \) and \([x]_B = \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}\).
\[ \vec{x} = 2 \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} + (-1) \begin{bmatrix} 3 \\ 2 \\ -2 \end{bmatrix} + 0 \begin{bmatrix} -4 \\ -7 \\ 0 \end{bmatrix} \]
\[ = \begin{bmatrix} (1)(2) + (3)(-1) + 4(0) \\ -2(2) + 2(-1) - 7(0) \\ 1(2) - 2(1) + 0(0) \end{bmatrix} = \begin{bmatrix} -1 \\ -6 \\ 4 \end{bmatrix} \]
This transcription provides the mathematical procedures for finding the \( B \)-coordinate vector and the vector \( x \) in relation to the given bases.
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