# Nonhomogeneous System: Similar Coefficients & Solutions?

• kosovo dave
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.
kosovo dave
Gold Member
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

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

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?

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

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

kosovo dave said:
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.

## 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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