Solution Found!
Continuing Exercise 3.2.10: Let A be an m n matrix. a. Use Theorem 2.5 to prove that
Chapter 3, Problem 24(choose chapter or problem)
Continuing Exercise 3.2.10: Let A be an \(m \times n\) matrix.
a. Use Theorem 2.5 to prove that \(\mathbf{N}\left(A^{\top} A\right)=\mathbf{N}(A)\). (Hint: If \(\mathbf{x} \in \mathbf{N}\left(A^{\top} A\right)\), then \(A \mathbf{x} \in \mathbf{C}(A) \cap \mathbf{N}\left(A^{\top}\right)\).)
b. Prove that \(\operatorname{rank}(A)=\operatorname{rank}\left(A^{\top} A\right)\).
c. Prove that \(\mathbf{C}\left(A^{\top} A\right)=\mathbf{C}\left(A^{\top}\right)\).
Questions & Answers
QUESTION:
Continuing Exercise 3.2.10: Let A be an \(m \times n\) matrix.
a. Use Theorem 2.5 to prove that \(\mathbf{N}\left(A^{\top} A\right)=\mathbf{N}(A)\). (Hint: If \(\mathbf{x} \in \mathbf{N}\left(A^{\top} A\right)\), then \(A \mathbf{x} \in \mathbf{C}(A) \cap \mathbf{N}\left(A^{\top}\right)\).)
b. Prove that \(\operatorname{rank}(A)=\operatorname{rank}\left(A^{\top} A\right)\).
c. Prove that \(\mathbf{C}\left(A^{\top} A\right)=\mathbf{C}\left(A^{\top}\right)\).
ANSWER:
Problem 24
Continuing Exercise 3.2.10: Let 
a. Use Theorem 2.5 to prove that 
(Hint: If 
b. Prove that 
c. Prove that 
Step by Step Solution
Step 1 of 3
(a)
Consider the given matrix,
Given that 
Then,
If 
So 
From equation (1) and (2), it is obtained that
