Nonhomogeneous System: Similar Coefficients & Solutions?

[kosovo dave]

This might belong in the HW section, but since it's specific to Linear Algebra I posted it here.

Alright, so we have a homogeneous system of 8 equations in 10 variables (an 8 x 10 matrix, let's call it A). We have found two solutions that are not multiples of each other (lets call them a and b), and every other solution is a linear combination of them. Can you be certain that any nonhomogeneous equation with the same coefficients has a solution?

I want to say yes, but I'm not sure why. Here's the stuff I know:
- Our solution for the homogeneous system is span{a, b}.
- Since there are free variables/the null space is not just 0 we know there are nontrivial solutions.
- Dim(Null(A))=8

 

[Simon Bridge]

Try constructing a non-homogeneous equation with those two coefficients that does not have a solution - consider that the inhomogeniety can be anything.

 

[kosovo dave]

I don't know why, but I'm still having a hard time with this :/ Could you give me another hint?

Here's what I tried:
Ax=c

if c= (0,0,1,0,0,0,0,0,0,0) I think the system would be inconsistent because row 3 of the augmented matrix would be all 0's and then a nonzero to the right of the vertical line. Does that work?

 

[Simon Bridge]

Well, how would you normally find the solution to a non-homogeneous system knowing the solution to the homogeneous one?

 

[kosovo dave]

augment the nonhomogeneous system with a solution from the homogeneous one?

 

[Stephen Tashi]

Can you be certain that any nonhomogeneous equation with the same coefficients has a solution?

For a matrix $A$ and a column vector $x$ the result of $Ax$ can be viewed as a linear combination of the column vectors of $A$ where the coefficients in the linear combination are the entries of $x$. So if $Ax = b$ has a solution, the vector $b$ must be in the span of the column vectors.