Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax = b. A= 14 -18 14 b= 10 4
Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax = b. A= 14 -18 14 b= 10 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Given:
\[
A = \begin{bmatrix}
1 & 4 \\
-1 & 8 \\
1 & 4
\end{bmatrix}, \quad
b = \begin{bmatrix}
10 \\
-4 \\
4
\end{bmatrix}
\]
Find:
(a) The orthogonal projection of vector \( \mathbf{b} \) onto the column space of matrix \( \mathbf{A} \).
(b) The least-squares solution of the equation \( \mathbf{A} \mathbf{x} = \mathbf{b} \).
**Solution Explanation:**
To solve these parts, follow these steps:
### (a) Orthogonal Projection of \( \mathbf{b} \) onto Col \( \mathbf{A} \)
1. **Formulate the Projection Formula:**
The orthogonal projection of vector \( \mathbf{b} \) onto the column space of \( \mathbf{A} \) (denoted as \( \text{Col} \mathbf{A} \)) can be represented as:
\[
\mathbf{P}_{\mathbf{b}} = \mathbf{A} (\mathbf{A}^T \mathbf{A})^{-1} \mathbf{A}^T \mathbf{b}
\]
2. **Calculate \( \mathbf{A}^T \):**
\[
\mathbf{A}^T = \begin{bmatrix}
1 & -1 & 1 \\
4 & 8 & 4
\end{bmatrix}
\]
3. **Compute \( \mathbf{A}^T \mathbf{A} \):**
\[
\mathbf{A}^T \mathbf{A} = \begin{bmatrix}
1 & -1 & 1 \\
4 & 8 & 4
\end{bmatrix}
\begin{bmatrix}
1 & 4 \\
-1 & 8 \\
1 & 4
\end{bmatrix} = \begin{bmatrix}
3 & 6 \\
6 & 96
\end{bmatrix}
\]
4. **Calculate the Inverse of \( \mathbf{A}^T \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5246d2a-5796-4784-a543-e596d3b5542c%2F7a36dc95-7715-43b3-bd42-b5702f7eb4d9%2Fbt9qjv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given:
\[
A = \begin{bmatrix}
1 & 4 \\
-1 & 8 \\
1 & 4
\end{bmatrix}, \quad
b = \begin{bmatrix}
10 \\
-4 \\
4
\end{bmatrix}
\]
Find:
(a) The orthogonal projection of vector \( \mathbf{b} \) onto the column space of matrix \( \mathbf{A} \).
(b) The least-squares solution of the equation \( \mathbf{A} \mathbf{x} = \mathbf{b} \).
**Solution Explanation:**
To solve these parts, follow these steps:
### (a) Orthogonal Projection of \( \mathbf{b} \) onto Col \( \mathbf{A} \)
1. **Formulate the Projection Formula:**
The orthogonal projection of vector \( \mathbf{b} \) onto the column space of \( \mathbf{A} \) (denoted as \( \text{Col} \mathbf{A} \)) can be represented as:
\[
\mathbf{P}_{\mathbf{b}} = \mathbf{A} (\mathbf{A}^T \mathbf{A})^{-1} \mathbf{A}^T \mathbf{b}
\]
2. **Calculate \( \mathbf{A}^T \):**
\[
\mathbf{A}^T = \begin{bmatrix}
1 & -1 & 1 \\
4 & 8 & 4
\end{bmatrix}
\]
3. **Compute \( \mathbf{A}^T \mathbf{A} \):**
\[
\mathbf{A}^T \mathbf{A} = \begin{bmatrix}
1 & -1 & 1 \\
4 & 8 & 4
\end{bmatrix}
\begin{bmatrix}
1 & 4 \\
-1 & 8 \\
1 & 4
\end{bmatrix} = \begin{bmatrix}
3 & 6 \\
6 & 96
\end{bmatrix}
\]
4. **Calculate the Inverse of \( \mathbf{A}^T \
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