The General Solution (HW #3)

The general (or ”complete” or ”total”) solution for the system A~x = ~b is the set of all solutions for thesystem.

Theorem: If ~xp is any solution to the system A~x = ~b, then the general solution for the system is the set~xp +N(A).

Proof: We must show that if A~x = ~b, then there exists ~n 2 N(A) so that ~x = ~xp + ~n.

But A(~x� ~xp) = ~0 so . . . finish the proof.

The tactics for solving A~x = ~b are then:

1) Use Gaussian Elimination on the augmented system.

2) Find a particular solution by setting the free variables to zero.

3) Find N(A) by replacing the reduced ~b with the zero vector.

4) Writing the final answer using the particular solution and a span of the ”special solutions.”

1

Find the general solution for A~x = ~b if A =

2

664

1 0 1 0 12 1 4 0 20 1 2 1 �11 0 1 �1 2

3

775 and ~b =

2

664

2961

3

775.

2

Find the general solution for B~x = ~b if B =

2

40 1 1 11 2 0 10 1 �1 0

3

5 and ~b =

2

4132

3

5.

3