Find a Basis for the solution set

  • Thread starter digitol87
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    Basis Set
In summary, the given equations can be rewritten as a matrix and reduced to show that they are essentially the same equation written three times. To solve the system, you can express each variable in terms of the others and see that the system is dependent, meaning any vector in the set can be written as a linear combination of two other vectors.
  • #1
digitol87
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3x1 + x2 + x3 = 0
6x1 + 2x2 + 2x3 = 0
-9x1 - 3x2 - 3x3 = 0


I'm not sure how to approach this problem. I've rewritten these equations as a matrix

[3 1 1]
[6 2 2]
[-9 -3 -3]

Reduced Echelon from gave me this
[3 1 1]
[0 0 0]
[0 0 0]

Am I approaching this the wrong way?
What should I do next? Please help.
Thank You.
 
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  • #2
digitol87 said:
3x1 + x2 + x3 = 0
6x1 + 2x2 + 2x3 = 0
-9x1 - 3x2 - 3x3 = 0
By inspection it can be seen that the 2nd equation is 2 times the first, and the 3rd is -3 times the first. In essence, you have the same equation written three times.
digitol87 said:
I'm not sure how to approach this problem. I've rewritten these equations as a matrix

[3 1 1]
[6 2 2]
[-9 -3 -3]

Reduced Echelon from gave me this
[3 1 1]
[0 0 0]
[0 0 0]

Am I approaching this the wrong way?
What should I do next? Please help.
Thank You.

Solve the first equation to get
x1 = -(1/3)x2 - (1/3)x3
x2 = x2 + 0x3
x3 = 0x2 + x3

The 2nd and 3rd equations above are obviously true.

If you stare at this system awhile, you might see that any vector <x1, x2, x3> in this set can be written as a linear combination of two vectors that happen to be linearly independent.
 

Related to Find a Basis for the solution set

1. What does it mean to find a basis for the solution set?

Finding a basis for the solution set means finding a set of vectors that can be used to represent all possible solutions to a system of equations. These vectors must be linearly independent and span the entire solution set.

2. When is it necessary to find a basis for the solution set?

It is necessary to find a basis for the solution set when solving a system of equations or finding the null space of a matrix. This allows for a more efficient and concise representation of all possible solutions.

3. How do you find a basis for the solution set?

To find a basis for the solution set, you can use the Gaussian elimination method to reduce the system of equations to its reduced row-echelon form. The pivot columns in the reduced matrix will correspond to the basis vectors for the solution set.

4. Can there be multiple bases for the same solution set?

Yes, there can be multiple bases for the same solution set. However, all bases for a given solution set will have the same number of vectors and will span the same subspace.

5. How is a basis for the solution set related to the dimension of the solution space?

The number of vectors in a basis for the solution set is equal to the dimension of the solution space. This means that the number of linearly independent vectors needed to represent all solutions to a system of equations is equal to the number of variables in the system.

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