Dimension of and Basis for Solution Set of Homogeneous System

• Oct 20th 2011, 05:04 PM
divinelogos
Dimension of and Basis for Solution Set of Homogeneous System
This is probably pretty simple I just want to make sure I'm doing it right:

Find the dimension of and a basis for the solution set of the following homogeneous systems of equations:

1. x1+2x2-3x3+x4=0

2.x1+2x2=0
x1-x2=0

Any help would be appreciated! Thanks! :)
• Oct 20th 2011, 11:40 PM
FernandoRevilla
Re: Dimension of and Basis for Solution Set of Homogeneous System
Quote:

Originally Posted by divinelogos
Find the dimension of and a basis for the solution set of the following homogeneous systems of equations: 1. x1+2x2-3x3+x4=0

Solving the system:

$\displaystyle \begin{bmatrix}{x_1}\\{x_2}\\{x_3}\\{x_4} \end{bmatrix}=\begin{bmatrix}{-2\lambda+2\mu-\gamma}\\{\lambda}\\{\mu}\\{\gamma}\end{bmatrix}= \lambda \begin{bmatrix}{-2}\\{1}\\{0}\\{0}\end{bmatrix}+\mu \begin{bmatrix}{2}\\{0}\\{1}\\{0}\end{bmatrix}+ \gamma \begin{bmatrix}{-1}\\{0}\\{0}\\{1}\end{bmatrix}$

Those three vectors span the subspace and are linearly independent. Try 2.