(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2Fe7619d79-cf74-4fde-9efe-8f648c297870%2F0xufzec_processed.jpeg&w=3840&q=75)
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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