Systems of Linear Equations

In summary, when solving for eigenvectors using substitution, the resulting systems of linear equations can have multiple solutions and the choice of which variable to use as the "free" variable does not affect the solution space. It is important to remember that the set of all eigenvectors corresponding to a given eigenvalue forms a vector space and can have multiple basis vectors.
  • #1
roam
1,271
12

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:

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

-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.
 
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  • #2
Both these are easy to solve by inspection. For example, in the first one, set x = 1 and y = -1.
 
  • #3
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.
 
  • #4
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 [itex]A-\lambda I[/itex] NOT have an inverse guarentees that your set of equation not have a single solution.

For the first, -3x-3y= 0, -4x- 4y= 0, it is not the case that "x,y are equal". Both equations are equivalent to -x-y= 0 and then y= -x, not x. Every eigenvector is of the form <x, -x> = x<1, -1>. The vector space of all eigenvectors is spanned by the single vector <1, -1>.

For the second set where every equation is of the form -x-y-z= 0 you have one equation in three variables. That means you can solve for any one in terms of the other two- it doesn't matter which you choose. For example, z= -x- y. Taking x=1, y= 0, z= -1 so <1, 0, -1> is in the space. Taking x= 0, y= 1, z= -1 so <0, 1, -1> is also in the space. Those two vectors form a basis for the space of eigenvectors.

If you had chosen instead to solve for y, y= -x- z. Now taking x=1, z= 0 gives <1, -1, 0> and taking x=0, z= 1 gives <0, -1, 1> . Those form a different basis for the same space.

If you had chosen to solve for x, x= -y- z giving <-1, 1, 0> and <-1, 0, 1>, yet another basis for the same space.
 

What is a system of linear equations?

A system of linear equations is a set of two or more equations that contain two or more variables. The solution to a system of linear equations is the set of values for the variables that satisfies all of the equations.

How do you solve a system of linear equations?

There are several methods for solving a system of linear equations, including substitution, elimination, and graphing. The most efficient method depends on the specific equations given in the system.

What is the difference between a consistent and an inconsistent system of linear equations?

A consistent system of linear equations has at least one solution that satisfies all of the equations. An inconsistent system has no solution, meaning that the equations are contradictory and cannot be satisfied simultaneously.

How do you graph a system of linear equations?

To graph a system of linear equations, you must first solve for one variable in each equation. Then, plot the resulting points on a coordinate plane and connect them to create a line. The point where the lines intersect is the solution to the system.

What real-world applications use systems of linear equations?

Systems of linear equations are used to model and solve various real-world problems, including mixture problems, cost and revenue analysis, and optimization problems. They are also used in fields such as engineering, economics, and physics.

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