# Linear algebra: nonhomogeneous system

## Homework Statement

Let A = $$\left( \begin{array}{l} \begin{array}{*{20}c} 0 & 1 & { - 1} \\ \end{array} \\ \begin{array}{*{20}c} 2 & 1 & 1 \\ \end{array} \\ \end{array} \right)$$. Suppose that for some b in $$\mathbb{R}^2$$, $$p = \left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ 1 \\ \end{array} } \right)$$ 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.

## Answers and Replies

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The solution is:

$$\left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ 1 \\ \end{array} } \right) + x_3 \left( {\begin{array}{*{20}c} { - 1} \\ 1 \\ 1 \\ \end{array} } \right),{\text{ }}x & _3 \in \mathbb{R}$$

But I can't make sense out of it.

Hurkyl
Staff Emeritus
Science Advisor
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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?

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.

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
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?

daniel_i_l
Gold Member
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?