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
