Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.4 Exercises - Page 41: 24

Answer

(a) True - According to Theorem 3, Ax = x_{1}a_{1} + x_{2}a_{2} + ... x_{n}a_{n}, where a_{1}... a^{n} are columns of A (b) True - Definition of Spanning Vectors (c) True - Ax = b --> x_{1}a_{1} + x_{2}a_{2} + ... x_{n}a_{n} = b which is the same solution set as [a_{1}, a_{2} ,..., a_{n}] (d) True - if Ax = b is inconsistent, b cannot be in the set spanned by the columns of A (e) Theorem 4 describes a coefficient matrix, not an augmented matrix; therefore, if [A b] has a pivot position in each row then it may or may not be consistent

Work Step by Step

(a) True - According to Theorem 3, Ax = x_{1}a_{1} + x_{2}a_{2} + ... x_{n}a_{n}, where a_{1}... a^{n} are columns of A (b) True - Definition of Spanning Vectors (c) True - Ax = b --> x_{1}a_{1} + x_{2}a_{2} + ... x_{n}a_{n} = b which is the same solution set as [a_{1}, a_{2} ,..., a_{n}] (d) True - if Ax = b is inconsistent, b cannot be in the set spanned by the columns of A (e) Theorem 4 describes a coefficient matrix, not an augmented matrix; therefore, if [A b] has a pivot position in each row then it may or may not be consistent
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