Find a basis for the solution space of the given homogeneous system.

In summary, in order to find the basis for the solution space of a homogeneous system, you must first reduce the matrix to reduced row echelon form. Then, the basis for the solution space is the column vector corresponding to the free variable in the reduced matrix.
  • #1
memo_juentes
8
0

Homework Statement


Find a basis for the solution space of the given homogeneous system.

x1 x2 x3 x4
1 2 -1 3 | 0
2 2 -1 6 | 0
1 0 0 3 | 0



The Attempt at a Solution


When I reduced to reduced row echelon form i get the following matrix:

1 0 0 3 | 0
0 1 0 0 | 0
0 0 1 0 | 0

Which I thought it meant that the basis for the solution space would be:

1
0
0
-3

But apparently it isn't...what am I doing wrong?
 
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  • #2
You didn't reduce the matrix correctly. Fix that first.
 
  • #3
I'm sorry I actually typed the matrix wrong when making this thread. The correct matrix is:

1 2 -1 3 |0
2 2 -1 6 |0
1 0 3 3 |0
 
  • #4
memo_juentes said:
When I reduced to reduced row echelon form i get the following matrix:

1 0 0 3 | 0
0 1 0 0 | 0
0 0 1 0 | 0

Which I thought it meant that the basis for the solution space would be:

1
0
0
-3

But apparently it isn't...what am I doing wrong?
Your reduced matrix says this:
x1 = -3x4
x2 = 0
x3 = 0
x4 = x4

This means that any vector x in the solution space is a constant multiple of what vector?
 
  • #5
Ohh got it, seems pretty obvious now that i see it

Thanks a lot btw
 
  • #6
You're welcome!
 

1. What is a homogeneous system?

A homogeneous system is a system of linear equations where all the constants on the right side of the equal sign are equal to zero.

2. What is a solution space?

A solution space is the set of all possible solutions to a system of equations. It can be represented as a vector space.

3. Why is finding a basis for the solution space important?

Finding a basis for the solution space allows us to represent the solution space in a more concise and organized manner. It also helps us to understand the structure and properties of the solution space.

4. How do you find a basis for the solution space of a homogeneous system?

To find a basis for the solution space, we first solve the system of equations and express the solutions in terms of parameters. Then, we use these parameters to create a set of vectors that span the solution space. Finally, we reduce this set of vectors to a linearly independent set, which forms the basis for the solution space.

5. Can a homogeneous system have more than one basis for its solution space?

Yes, a homogeneous system can have multiple bases for its solution space. This is because there can be different ways to express the solutions in terms of parameters, resulting in different sets of spanning vectors. However, all of these bases will have the same number of vectors and will span the same solution space.

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