Show that the given basis for S is orthogonal. 1 4 S = span S = 6- Let s1 = and s2 = Then s1· S2 = so {S1, S2} ---?--- 8 an orthogonal basis for S. 6. Write s as a linear combination of the basis vectors. (Give your answer in terms of s, and s2.) S =
Show that the given basis for S is orthogonal. 1 4 S = span S = 6- Let s1 = and s2 = Then s1· S2 = so {S1, S2} ---?--- 8 an orthogonal basis for S. 6. Write s as a linear combination of the basis vectors. (Give your answer in terms of s, and s2.) S =
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
![**Show that the given basis for \( S \) is orthogonal.**
\[ S = \text{span} \left( \left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right], \left[ \begin{array}{c} 1 \\ -1 \\ 6 \end{array} \right] \right), \quad \mathbf{s} = \left[ \begin{array}{c} 1 \\ 4 \\ -9 \end{array} \right] \]
Let
\[
\mathbf{s}_1 = \left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right] \quad \text{and} \quad \mathbf{s}_2 = \left[ \begin{array}{c} 1 \\ -1 \\ 6 \end{array} \right]
\]
Then
\[
\mathbf{s}_1 \cdot \mathbf{s}_2 = \underline{\qquad}
\]
so \(\{\mathbf{s}_1, \mathbf{s}_2\} \) is an orthogonal basis for \( S \).
**Write \(\mathbf{s}\) as a linear combination of the basis vectors. (Give your answer in terms of \(\mathbf{s}_1\) and \(\mathbf{s}_2\).)**
\[ \mathbf{s} = \underline{\qquad} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef36e706-47e8-4994-a8d4-317113440949%2Fa3492220-f166-4413-b3eb-ffeb7c38a58e%2Fqbv8n0g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Show that the given basis for \( S \) is orthogonal.**
\[ S = \text{span} \left( \left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right], \left[ \begin{array}{c} 1 \\ -1 \\ 6 \end{array} \right] \right), \quad \mathbf{s} = \left[ \begin{array}{c} 1 \\ 4 \\ -9 \end{array} \right] \]
Let
\[
\mathbf{s}_1 = \left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right] \quad \text{and} \quad \mathbf{s}_2 = \left[ \begin{array}{c} 1 \\ -1 \\ 6 \end{array} \right]
\]
Then
\[
\mathbf{s}_1 \cdot \mathbf{s}_2 = \underline{\qquad}
\]
so \(\{\mathbf{s}_1, \mathbf{s}_2\} \) is an orthogonal basis for \( S \).
**Write \(\mathbf{s}\) as a linear combination of the basis vectors. (Give your answer in terms of \(\mathbf{s}_1\) and \(\mathbf{s}_2\).)**
\[ \mathbf{s} = \underline{\qquad} \]
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