Prove Theorem 3 as follows: Given an m × n matrix A, an

QUESTION:

Problem 29E

Prove Theorem 3 as follows: Given an m × n matrix A, an element in Col A has the form Ax for some x in . Let Ax and Aw represent any two vectors in Col A.

a. Explain why the zero vector is in Col A.

b. Show that the vector Ax + Aw is in Col A.

c. Given a scalar c, show that c (Ax) is in Col A.

Theorem 3: The column space of an m × n matrix A is a subspace of .