In Exercises 11 and 12, the matrices are all n × n. Each part of the exercises is an implication of the form “If 〈 statement 1〉, then 〈 statement 2 〉.” Mark an implication as True if the truth of 〈 statement 2 〉 always follows whenever 〈 statement 1 〉 happens to be true. An implication is False if there is an instance in which 〈 statement 2 〉 is false but 〈 statement 1 〉 is true. Justify each answer.
a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n × n identity matrix.
b. If the columns of A span ℝn, then the columns are linearly independent.
c. If A is an n × n matrix, then the equation Ax = b has at least one solution for each b in ℝn.
d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions.
e. If AT is not invertible, then A is not invertible.
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