**Exercise 1:**(a) Let \( A \in M_{m \times n}(\mathbb{F}) \) and let \( A_{\text{re}} \) be the reduced echelon form of \( A \). Prove that \( Ax = 0 \) if and only if \( A_{\text{re}}x = 0 \).(b) Suppose \( A \in M_{3 \times 4}(\mathbb{R}) \) is such that \( Ax = 0 \) has the two solutions:\[ \mathbf{x_1} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} \quad \text{and} \quad \mathbf{x_2} = \begin{bmatrix} -2 \\ -1 \\ 0 \\ 1 \end{bmatrix} \]and any other solution to \( Ax = 0 \) is a linear combination of \( \mathbf{x_1} \) and \( \mathbf{x_2} \).i. Find the reduced echelon form \( A_{\text{re}} \) of \( A \). Make sure to explain/justify your answer!ii. Find the dimensions of the four fundamental subspaces of \( A \).

Answered: Image /qna-images/answer/e7619d79-cf74-4fde-9efe-8f648c297870.jpgThat completes the solution. Hope this helps. Kindly upvote if you find this helpful. Thank you.

Answered: (a) Let A € Mmxn (F) and let Are be the…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 51EQ

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