# 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.

Related Calculus and Beyond Homework Help News on Phys.org
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
Gold Member
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