Nonhomogeneous System: Similar Coefficients & Solutions?

In summary, if there is a vector b that is a linear combination of the column vectors of A and has a solution, then b must be in the span of the column vectors.
  • #1
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
 
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  • #2
Try constructing a non-homogeneous equation with those two coefficients that does not have a solution - consider that the inhomogeniety can be anything.
 
  • #3
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?
 
  • #4
Well, how would you normally find the solution to a non-homogeneous system knowing the solution to the homogeneous one?
 
  • #5
augment the nonhomogeneous system with a solution from the homogeneous one?
 
  • #6
kosovo dave said:
Can you be certain that any nonhomogeneous equation with the same coefficients has a solution?
For a matrix [itex] A [/itex] and a column vector [itex] x [/itex] the result of [itex] Ax [/itex] can be viewed as a linear combination of the column vectors of [itex] A [/itex] where the coefficients in the linear combination are the entries of [itex] x [/itex]. So if [itex] Ax = b [/itex] has a solution, the vector [itex] b [/itex] must be in the span of the column vectors.
 

Related to Nonhomogeneous System: Similar Coefficients & Solutions?

1. What is a nonhomogeneous system with similar coefficients?

A nonhomogeneous system is a set of linear equations with different variables and constants on one side and equal to a constant on the other side. Similar coefficients refer to the fact that the coefficients of each variable in the system are the same in every equation.

2. How are nonhomogeneous systems with similar coefficients solved?

Nonhomogeneous systems with similar coefficients can be solved using the method of undetermined coefficients, where a particular solution is found by guessing the form of the solution and then solving for the unknown coefficients.

3. Can a nonhomogeneous system with similar coefficients have multiple solutions?

Yes, a nonhomogeneous system with similar coefficients can have multiple solutions. This can occur when the system is underdetermined, meaning there are more variables than equations, or when there is a free variable that can take on different values.

4. What is the difference between a homogeneous and nonhomogeneous system with similar coefficients?

A homogeneous system has all constant terms on the right-hand side equal to zero, while a nonhomogeneous system has at least one non-zero constant term. Additionally, a homogeneous system always has the trivial solution (all variables equal to 0), while a nonhomogeneous system may or may not have a trivial solution.

5. Can a nonhomogeneous system with similar coefficients be solved using Gaussian elimination?

Yes, Gaussian elimination can be used to solve a nonhomogeneous system with similar coefficients. However, it may not be the most efficient method as it requires finding the inverse of a matrix, which can be time-consuming for larger systems. The method of undetermined coefficients may be a better approach for solving these types of systems.

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