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Suppose that c¹, c² are both 3 × 1 matrices. Let A = [c¹c²] be augmented matrix with c¹ and c² and c³ as its columns. Answer the following (each answer should be a short sentence or two.) a.) What are the dimensions of A? (That is, how many rows and columns does it have?) b.) Does the equation Ax = b always have to have a solution? c.) Based on your answer to the last part, does (c¹, c²) have to include every 3 × 1 matrix? d.) Can the equation Ar = 0 have multiple solutions? e.) Based on your answer to the last part, do the vectors {c¹, (²) have to be linearly inde- pendent?

Suppose that c¹, c² are both 3 × 1 matrices. Let A = [c¹c²] be augmented matrix with c¹ and c² and c³ as its columns. Answer the following (each answer should be a short sentence or two.) a.) What are the dimensions of A? (That is, how many rows and columns does it have?) b.) Does the equation Ax = b always have to have a solution? c.) Based on your answer to the last part, does (c¹, c²) have to include every 3 × 1 matrix? d.) Can the equation Ar = 0 have multiple solutions? e.) Based on your answer to the last part, do the vectors {c¹, (²) have to be linearly inde- pendent?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 42EQ
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Suppose that c¹, c² are both 3 × 1 matrices. Let A = [c¹c²] be augmented matrix with c¹ and c² and c³ as its columns. Answer the following (each answer should be a short sentence or two.) a.) What are the dimensions of A? (That is, how many rows and columns does it have?) b.) Does the equation Ax = b always have to have a solution? c.) Based on your answer to the last part, does (c¹, c²) have to include every 3 × 1 matrix? d.) Can the equation Ar = 0 have multiple solutions? e.) Based on your answer to the last part, do the vectors {c¹, (²) have to be linearly inde- pendent?
Transcribed Image Text:Suppose that c¹, c² are both 3 × 1 matrices. Let A = [c¹c²] be augmented matrix with c¹ and c² and c³ as its columns. Answer the following (each answer should be a short sentence or two.) a.) What are the dimensions of A? (That is, how many rows and columns does it have?) b.) Does the equation Ax = b always have to have a solution? c.) Based on your answer to the last part, does (c¹, c²) have to include every 3 × 1 matrix? d.) Can the equation Ar = 0 have multiple solutions? e.) Based on your answer to the last part, do the vectors {c¹, (²) have to be linearly inde- pendent?
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