(2) Let A € Rmxn and let à be a solution of the least squares problem Az = b. Show that a vector y ER" will also be a solution if and only if y = 2 + z, for some vector z € N(A). [HINT: N(ATA) = N(A)].
(2) Let A € Rmxn and let à be a solution of the least squares problem Az = b. Show that a vector y ER" will also be a solution if and only if y = 2 + z, for some vector z € N(A). [HINT: N(ATA) = N(A)].
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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2.
![**Problem Statement:**
Let \( A \in \mathbb{R}^{m \times n} \) and let \( \hat{x} \) be a solution of the least squares problem \( Ax = b \). Show that a vector \( y \in \mathbb{R}^n \) will also be a solution if and only if \( y = \hat{x} + z \), for some vector \( z \in N(A) \). [HINT: \( N(A^T A) = N(A) \)].
**Detailed Explanation:**
- **Vector Spaces:**
- \( \mathbb{R}^{m \times n} \): The space of all \( m \times n \) matrices with real numbers.
- \( \mathbb{R}^n \): The space of all \( n \)-dimensional vectors with real numbers.
- \( N(A) \): The null space of the matrix \( A \), which is the set of all vectors \( x \) such that \( Ax = 0 \).
- \( N(A^T A) \): The null space of the matrix \( A^T A \), where \( A^T \) is the transpose of \( A \).
- **Least Squares Problem:**
- The least squares problem seeks to find a vector \( \hat{x} \) such that \( Ax \approx b \), minimizing the residual \( \|Ax - b\| \).
- **Solution Characterization:**
- The problem asks to show that any vector \( y \) that is also a solution to the least squares problem can be written as \( \hat{x} + z \), where \( z \) is in the null space of \( A \).
- **Hint Utilization:**
- The hint provided states that the null space of \( A^T A \) is the same as the null space of \( A \), i.e., \( N(A^T A) = N(A) \).
**Steps to Solution:**
1. **Understanding Null Spaces:**
- The null space \( N(A) \) consists of all vectors \( z \) such that \( Az = 0 \).
- If \( y \) is another solution, \( A\hat{x} ≈ b \) and \( Ay ≈ b \).
2. **Formulating y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13d3a07d-24ab-4d07-ab36-ffae21b1955b%2F915c0789-0b1e-4afa-ae50-0deb4898e892%2Fmyjafg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( A \in \mathbb{R}^{m \times n} \) and let \( \hat{x} \) be a solution of the least squares problem \( Ax = b \). Show that a vector \( y \in \mathbb{R}^n \) will also be a solution if and only if \( y = \hat{x} + z \), for some vector \( z \in N(A) \). [HINT: \( N(A^T A) = N(A) \)].
**Detailed Explanation:**
- **Vector Spaces:**
- \( \mathbb{R}^{m \times n} \): The space of all \( m \times n \) matrices with real numbers.
- \( \mathbb{R}^n \): The space of all \( n \)-dimensional vectors with real numbers.
- \( N(A) \): The null space of the matrix \( A \), which is the set of all vectors \( x \) such that \( Ax = 0 \).
- \( N(A^T A) \): The null space of the matrix \( A^T A \), where \( A^T \) is the transpose of \( A \).
- **Least Squares Problem:**
- The least squares problem seeks to find a vector \( \hat{x} \) such that \( Ax \approx b \), minimizing the residual \( \|Ax - b\| \).
- **Solution Characterization:**
- The problem asks to show that any vector \( y \) that is also a solution to the least squares problem can be written as \( \hat{x} + z \), where \( z \) is in the null space of \( A \).
- **Hint Utilization:**
- The hint provided states that the null space of \( A^T A \) is the same as the null space of \( A \), i.e., \( N(A^T A) = N(A) \).
**Steps to Solution:**
1. **Understanding Null Spaces:**
- The null space \( N(A) \) consists of all vectors \( z \) such that \( Az = 0 \).
- If \( y \) is another solution, \( A\hat{x} ≈ b \) and \( Ay ≈ b \).
2. **Formulating y
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