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(a) Let A € Mmxn (F) and let Are be the reduced echelon form of A. Prove that Ax = 0 if and only if Arex = 0. (b) Suppose A € M3x4 (R) is such that Ax = 0 has the two solutions --0- X1 = and X2 and any other solution to Ax = 0 is a linear combination of x₁ and x2. i. Find the reduced echelon form Are of A. Make sure to explain/justify your answer! ii. Find the dimensions of the four fundamental subspaces of A.

(a) Let A € Mmxn (F) and let Are be the reduced echelon form of A. Prove that Ax = 0 if and only if Arex = 0. (b) Suppose A € M3x4 (R) is such that Ax = 0 has the two solutions --0- X1 = and X2 and any other solution to Ax = 0 is a linear combination of x₁ and x2. i. Find the reduced echelon form Are of A. Make sure to explain/justify your answer! ii. Find the dimensions of the four fundamental subspaces of A.

Linear Algebra: A Modern Introduction
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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**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 \( A \).
Transcribed Image Text:**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 \( A \).
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