# Systems of Linear Equations

## Homework Statement

For a few problems dealing with eigenvectors, I substituted my eigenvalues into the characteristic equations. I got systems of linear equations. I need to find the general solution to the systems in order to find the corresponding eigenvectors.
For example in one problem I have to solve:

$$A = \left[\begin{array}{ccccc} -3&-3 \\ -4&-4 \end{array}\right]$$ $$\left[\begin{array}{ccccc} x\\ y \end{array}\right]$$ $$= \left[\begin{array}{ccccc} 0\\ 0 \end{array}\right]$$

-3x-3y=0
-4x-4y=0

It looks as if x,y are equal. I think I might need to write one in terms of the other but I'm no sure which.

or for example the system of equations:

-x-y-z=0
-x-y-z=0
-x-y-z=0

We have 3 variables and 3 equations which are exactly the same. How do we decide which variable to use as the "free" variable?

Thanks.

dx
Homework Helper
Gold Member
Both these are easy to solve by inspection. For example, in the first one, set x = 1 and y = -1.

Pengwuino
Gold Member
It doesn't matter. In the 2nd example, you can pick x = -y-z or y = -x-z or z = -x-y. You will still have the same solution space even though the eigenvectors don't look the same.

HallsofIvy
Remember that the set of all eigenvectors corresponding to a given eigenvalue for a vector space- there is not one single solution. In fact, the condition that $A-\lambda I$ NOT have an inverse guarentees that your set of equation not have a single solution.