Linear algebra: nonhomogeneous system

In summary, we are given a nonhomogeneous equation Ax = b with a particular solution p. The general form of the solution to this equation is p + x3v, where x3 is any real number and v is a vector in the solution space of the homogeneous equation Ax = 0. To find the solutions to the nonhomogeneous equation, we can use the particular solution p and add it to any solution from the solution space of the homogeneous equation.
  • #1
494
2

Homework Statement


Let A = [tex]\left( \begin{array}{l} \begin{array}{*{20}c} 0 & 1 & { - 1} \\ \end{array} \\ \begin{array}{*{20}c} 2 & 1 & 1 \\ \end{array} \\
\end{array} \right)[/tex]. Suppose that for some b in [tex]\mathbb{R}^2[/tex], [tex]p = \left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ 1 \\ \end{array} } \right)[/tex] is one particular solution to the nonhomogeneous equation Ax = b. What is the general form of the solution to this equation (Ax = b)?


Homework Equations


A relevant definition: A nonhomogeneous system is one with Ax = b, b is not equal to zero. A is a matrix and x and b are vectors.


The Attempt at a Solution


Since a homogeneous equation is Ax = 0, b = 0 in this particular question. Given p, I can presume that the general form of the solution will include p, but I'm not sure how to proceed.
 
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  • #2
The solution is:

[tex]\left( {\begin{array}{*{20}c}
1 \\
{ - 1} \\
1 \\

\end{array} } \right) + x_3 \left( {\begin{array}{*{20}c}
{ - 1} \\
1 \\
1 \\

\end{array} } \right),{\text{ }}x & _3 \in \mathbb{R}[/tex]

But I can't make sense out of it.
 
  • #3
Well, first off you can verify that actually is a family of solutions, right?

You seem to think the homogeneous equation Ax=0 has some bearing; well, what is the solution space to Ax=0?
 
  • #4
I'm guessing since it's in the solution, the solution space is contained in R, but I'm not sure how to verify that it is actually a family of solutions. I'm trying to understand the presentation of the solution in terms of two vectors, one being multiplied by x3 for some reason.
 
  • #5
z-component said:
I'm not sure how to verify that it is actually a family of solutions.
Multiply it (on the left) by A. What do you get?
 
  • #6
Given one solution and all the solutions to the respective homogeneous equation how do you find the solutions to the non-homogeneous equation?
HINT: what happens if you add the two equations together?
 

What is a nonhomogeneous system in linear algebra?

A nonhomogeneous system in linear algebra is a system of linear equations where the constant terms are not all equal to zero. This means that there is not a unique solution to the system.

How do you solve a nonhomogeneous system in linear algebra?

To solve a nonhomogeneous system in linear algebra, you can use various methods such as Gaussian elimination, Cramer's rule, or matrix inversion. These methods involve manipulating the equations to isolate the variables and find a solution that satisfies all of the equations.

Can a nonhomogeneous system have no solutions?

Yes, a nonhomogeneous system can have no solutions. This occurs when the equations are inconsistent and cannot be satisfied simultaneously.

What is the difference between a homogeneous and a nonhomogeneous system?

The main difference between a homogeneous and a nonhomogeneous system is in the constant terms. In a homogeneous system, all constant terms are equal to zero, while in a nonhomogeneous system, at least one constant term is non-zero.

What are some real-world applications of nonhomogeneous systems in linear algebra?

Nonhomogeneous systems in linear algebra have various real-world applications, including in engineering, economics, and physics. For example, they can be used to model electrical circuits, chemical reactions, and economic markets.

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