Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Problem Description:**
Determine the orthogonal projection of the vector
\[
\begin{bmatrix}
1 \\
-1 \\
3 \\
2
\end{bmatrix}
\]
onto the subspace \( V \) spanned by the vectors
\[
\begin{bmatrix}
2 \\
1 \\
2 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
1 \\
-2 \\
1
\end{bmatrix}
\].
**Options:**
A.
\[
\begin{bmatrix}
10/9 \\
7/9 \\
2 \\
-2/9
\end{bmatrix}
\]
B.
\[
\begin{bmatrix}
-5/9 \\
14/9 \\
7/9 \\
1
\end{bmatrix}
\]
C.
\[
\begin{bmatrix}
5/9 \\
11/9 \\
-7/9 \\
3
\end{bmatrix}
\]
D.
\[
\begin{bmatrix}
8/9 \\
-7/9 \\
2/9 \\
3
\end{bmatrix}
\]
**Explanation:**
To find the orthogonal projection of a vector onto a subspace spanned by multiple vectors, follow these steps:
1. Construct a matrix with the spanning vectors as columns.
2. Use the formula for projection: if \( A \) is the matrix of the spanning vectors, then the projection of a vector \( \mathbf{b} \) onto the column space of \( A \) is given by:
\[
\text{proj}_{\text{Col}(A)}(\mathbf{b}) = A(A^TA)^{-1}A^T\mathbf{b}
\]
3. Compute this expression to obtain the projected vector, then compare it with the provided options.
This procedure helps in determining the option that contains the vector representing the orthogonal projection of the given vector onto the specified subspace.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0114c863-1109-486a-b8d9-8254bb1b16b5%2Fb5809697-49ee-4420-8804-2e1dd1b9b197%2Ffdfwq9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Determine the orthogonal projection of the vector
\[
\begin{bmatrix}
1 \\
-1 \\
3 \\
2
\end{bmatrix}
\]
onto the subspace \( V \) spanned by the vectors
\[
\begin{bmatrix}
2 \\
1 \\
2 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
1 \\
-2 \\
1
\end{bmatrix}
\].
**Options:**
A.
\[
\begin{bmatrix}
10/9 \\
7/9 \\
2 \\
-2/9
\end{bmatrix}
\]
B.
\[
\begin{bmatrix}
-5/9 \\
14/9 \\
7/9 \\
1
\end{bmatrix}
\]
C.
\[
\begin{bmatrix}
5/9 \\
11/9 \\
-7/9 \\
3
\end{bmatrix}
\]
D.
\[
\begin{bmatrix}
8/9 \\
-7/9 \\
2/9 \\
3
\end{bmatrix}
\]
**Explanation:**
To find the orthogonal projection of a vector onto a subspace spanned by multiple vectors, follow these steps:
1. Construct a matrix with the spanning vectors as columns.
2. Use the formula for projection: if \( A \) is the matrix of the spanning vectors, then the projection of a vector \( \mathbf{b} \) onto the column space of \( A \) is given by:
\[
\text{proj}_{\text{Col}(A)}(\mathbf{b}) = A(A^TA)^{-1}A^T\mathbf{b}
\]
3. Compute this expression to obtain the projected vector, then compare it with the provided options.
This procedure helps in determining the option that contains the vector representing the orthogonal projection of the given vector onto the specified subspace.
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