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kosovo dave
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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
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