**10.10.** Let \( W \) be the subspace of \( \mathbb{R}^4 \) given by: \[ W = \text{Span} \left\{ \begin{bmatrix} 1 \\ -3 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 7 \\ 0 \\ 2 \end{bmatrix} \right\}. \] Find a basis for \(W^{\perp}\), the orthogonal complement of \( W \). ### 10.10 Basis for \(W^\perp\) is \[ \left\{ \begin{bmatrix} 14 \\ 4 \\ -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}. \] #### Explanation: The expression denotes that the basis for the orthogonal complement \(W^\perp\) consists of the set containing two vectors. - The first vector in the set is: \[ \begin{bmatrix} 14 \\ 4 \\ -1 \\ 0 \end{bmatrix} \] - The second vector in the set is: \[ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \] These vectors form a basis for the subspace \(W^\perp\), meaning they are linearly independent and span \(W^\perp\).
**10.10.** Let \( W \) be the subspace of \( \mathbb{R}^4 \) given by: \[ W = \text{Span} \left\{ \begin{bmatrix} 1 \\ -3 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 7 \\ 0 \\ 2 \end{bmatrix} \right\}. \] Find a basis for \(W^{\perp}\), the orthogonal complement of \( W \). ### 10.10 Basis for \(W^\perp\) is \[ \left\{ \begin{bmatrix} 14 \\ 4 \\ -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}. \] #### Explanation: The expression denotes that the basis for the orthogonal complement \(W^\perp\) consists of the set containing two vectors. - The first vector in the set is: \[ \begin{bmatrix} 14 \\ 4 \\ -1 \\ 0 \end{bmatrix} \] - The second vector in the set is: \[ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \] These vectors form a basis for the subspace \(W^\perp\), meaning they are linearly independent and span \(W^\perp\).
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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![**10.10.** Let \( W \) be the subspace of \( \mathbb{R}^4 \) given by:
\[ W = \text{Span} \left\{ \begin{bmatrix}
1 \\
-3 \\
2 \\
-1
\end{bmatrix},
\begin{bmatrix}
-2 \\
7 \\
0 \\
2
\end{bmatrix} \right\}. \]
Find a basis for \(W^{\perp}\), the orthogonal complement of \( W \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8efe9e31-ffe4-44e3-b161-05dc2bae8f47%2F97f48342-fcb7-4f7d-a0dd-fb10a5b5abb2%2Fdko3pi7_processed.png&w=3840&q=75)
Transcribed Image Text:**10.10.** Let \( W \) be the subspace of \( \mathbb{R}^4 \) given by:
\[ W = \text{Span} \left\{ \begin{bmatrix}
1 \\
-3 \\
2 \\
-1
\end{bmatrix},
\begin{bmatrix}
-2 \\
7 \\
0 \\
2
\end{bmatrix} \right\}. \]
Find a basis for \(W^{\perp}\), the orthogonal complement of \( W \).
![### 10.10 Basis for \(W^\perp\) is
\[
\left\{
\begin{bmatrix}
14 \\
4 \\
-1 \\
0
\end{bmatrix},
\begin{bmatrix}
1 \\
0 \\
0 \\
1
\end{bmatrix}
\right\}.
\]
#### Explanation:
The expression denotes that the basis for the orthogonal complement \(W^\perp\) consists of the set containing two vectors.
- The first vector in the set is:
\[
\begin{bmatrix}
14 \\
4 \\
-1 \\
0
\end{bmatrix}
\]
- The second vector in the set is:
\[
\begin{bmatrix}
1 \\
0 \\
0 \\
1
\end{bmatrix}
\]
These vectors form a basis for the subspace \(W^\perp\), meaning they are linearly independent and span \(W^\perp\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8efe9e31-ffe4-44e3-b161-05dc2bae8f47%2F97f48342-fcb7-4f7d-a0dd-fb10a5b5abb2%2Flebqna_processed.png&w=3840&q=75)
Transcribed Image Text:### 10.10 Basis for \(W^\perp\) is
\[
\left\{
\begin{bmatrix}
14 \\
4 \\
-1 \\
0
\end{bmatrix},
\begin{bmatrix}
1 \\
0 \\
0 \\
1
\end{bmatrix}
\right\}.
\]
#### Explanation:
The expression denotes that the basis for the orthogonal complement \(W^\perp\) consists of the set containing two vectors.
- The first vector in the set is:
\[
\begin{bmatrix}
14 \\
4 \\
-1 \\
0
\end{bmatrix}
\]
- The second vector in the set is:
\[
\begin{bmatrix}
1 \\
0 \\
0 \\
1
\end{bmatrix}
\]
These vectors form a basis for the subspace \(W^\perp\), meaning they are linearly independent and span \(W^\perp\).
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