• Support PF! Buy your school textbooks, materials and every day products Here!

Linear algebra: nonhomogeneous system

  • #1
478
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.
 

Answers and Replies

  • #2
478
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
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
18
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
478
2
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
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
18
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
daniel_i_l
Gold Member
867
0
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?
 

Related Threads for: Linear algebra: nonhomogeneous system

Replies
3
Views
2K
Replies
1
Views
846
  • Last Post
Replies
7
Views
2K
Replies
3
Views
2K
  • Last Post
Replies
5
Views
578
Replies
9
Views
1K
Replies
0
Views
3K
Replies
2
Views
482
Top