Question

7. Answer the following questions true or false and provide an explanation. • If you think the statement is true, refer to a definition or theorem. • If false, give a counter-example to show that the statement is not true for all cases.

(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then the equation Ax = b has a unique solution for all b in R^3 .

(b) If the augmented matrix [ A b ] has a non-pivot column then the system Ax = b has infinitely many solutions.

(c) If a1 and a2 are vectors in R^3 such that a1 is not a scalar multiple of a2, then {a1, a2} is an linearly independent set.

(d) Let A be a 6 × 4 matrix. If the columns of A are linearly independent then the columns of A span R^6 .

(e) Let T : R^3 → R^3 be a linear transformation. If T is onto, then T must also be one-to-one.

(f) Let a1, a2 and a3 be vectors in R^3 . If there exists real numbers x1, x2, x3 such that x1a1 + x2a2 + x3a3 = 0 then {a1, a2, a3} is a linearly dependent set.

Answer 1