# Solutions of Linear Systems III

• May 5th 2011, 05:55 PM
soulkeeper
Solutions of Linear Systems III
Write a short paragraph to answer each of the following questions about solutions of systems of linear equations. You should not perform any calculations, but instead base your explanations on the appropriate properties.

[5] The linear systems Ax = b1 + b2 necessarily consistent? Why or why not?

[6] Consider the linear system Ax = b. If the rank of A equals the rank of the augmented matrix for the system, explain why the system must be consistent. Contrast this to the case in which the rank of A is less than the rank of the augmented matrix.
• May 6th 2011, 07:17 AM
HappyJoe
As for [5], see if you can come up with an example, where the linear system Ax = b1 + b2 is inconsistent. For starters, let b2 be the zero vector and think about cases where A is 2x2.

For [6], the rank of A is the maximum number of linearly independent columns of A. If the augmented matrix has the same rank, then the maximum number of linearly independent columns of the augmented system is the same as the maximum number of linearly independent columns of A. What does this tell you about the relation between the vector b and the columns of A?