Linear Systems - Mathematics 214 - Fall 2007

1. Let A be an m x n matrix. In class we said that the leading columns of A form a basis for col(A). But we only discussed the fact that the leading columns of A are linearly independent --- we forgot to discuss why they span col(A). This is what you'll prove in this problem:
   1. Explain why in rref(A) every free column is a linear combination of the leading columns.
   2. Explain why in A every free column is a linear combination of the leading columns.
   3. Prove that if a vector is a linear combination of all the columns of A, then it is a linear combination of the leading columns of A. Hint: Use part b above plus "The HW Problem" (Section 2.3 problem 21).
   4. Prove that the leading columns of A span col(A).
2. Assume A and B are invertible n x n matrices. Determine whether each of the following proofs is correct or incorrect:
   

   Proof that (AB)-1 = B-1A-1	Proof that (AB)-1 = A-1B-1	Proof that 1 = 2
   (AB)-1 = B-1A-1	(AB)-1 = A-1B-1	1 = 2
   ((AB)-1)-1 = (B-1A-1)-1	((AB)-1)-1 = (A-1B-1)-1	0(1) = 0(2)
   AB = (A-1)-1( B-1)-1	AB = (A-1 )-1( B-1)-1	0 = 0
   AB = AB	AB = AB