Math Help - Linear Algebra Problem

1. Linear Algebra Problem

Hey, I have this linear alfebra problem that I'm stuck on. here's the actual problem:

Consider a homogenous system of 7 linear equations in 9 variables. Suppose that there are two solutions to the system that are not multiples of one another, and all other solutions are linear combinations of these solutions. Will the associated nonhomogeneous system have a solution for every possible choice of constants on the right sides of the equations? Justify answer carefully.

Now, I know that I'm dealing with a 7x9 matrix, but that's where it ends. I spent alot of time trying to undersand, but it's not working for me.

Any insight would be GREAT.

2. Originally Posted by Skinner
Hey, I have this linear alfebra problem that I'm stuck on. here's the actual problem:

Consider a homogenous system of 7 linear equations in 9 variables. Suppose that there are two solutions to the system that are not multiples of one another, and all other solutions are linear combinations of these solutions. Will the associated nonhomogeneous system have a solution for every possible choice of constants on the right sides of the equations? Justify answer carefully.

Now, I know that I'm dealing with a 7x9 matrix, but that's where it ends. I spent alot of time trying to undersand, but it's not working for me.

Any insight would be GREAT.
the answer is yes! here's why:

let A be the $7 \times 9$ matrix. what you're given about the solutions of the homogeneous system means that the nullity of A is 2. so by the rank-nullity theorem: rank(A) = 9 - nullity(A) = 7. Q.E.D.

3. Yes, what noncommalg says is based on the following useful face:

n=rank(A) +dim(Nul(A))

and rank(A) = dim(Col(A))

and dim(Nul(A)) are the number of free variables in your homogenous solution.

4. I get it now! Since there's a pivot position in every row of the given matrix, there exists a solution for any $b$ in the nonhomogeneous equation $Ax=b$.

Thanks for the help!