Algebra, the basis of a solution space

In summary, the basis for the solution space W is <1, -1, 0, -1/3> and <0, -3, 1, 1/3>. This basis spans W and is independent.
  • #1
FunkReverend
5
0

Homework Statement


Find the basis of the solution space W [itex]\subset[/itex] [itex]\Re^{4}[/itex]
of the system of linear equations

[itex]2x_{1}[/itex] + [itex]1x_{2}[/itex] + [itex]2x_{3}[/itex] +[itex]3x_{4}[/itex] =0
[itex] _{ }[/itex]
[itex]1x_{1}[/itex] + [itex]1x_{2}[/itex] + [itex]3x_{3}[/itex] = 0


Homework Equations


The basis must span W and be independent.


The Attempt at a Solution


Solving the above system, I get
[itex]x_{2}[/itex] = [itex]-x_{1}[/itex] - [itex]x_{3}[/itex]
[itex]x_{4}[/itex] = [itex]\frac{x_{3}-x_{1}}{3}[/itex]

With 2 degrees of freedom, [itex]x_{1}[/itex] and [itex]x_{3}[/itex],
so I must need a 2D basis. I separately fixed [itex]x_{1}[/itex] and [itex]x_{3}[/itex] to 1 and the other to zero and got the following vectors:
[1, -1, 0, -1/3] and [0, -3, 1, 1/3]
I feel like this is right, as I've been looking up some examples, but I'm not sure this spans all the solutions.

Am I on the right track?
 
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  • #2
FunkReverend said:

Homework Statement


Find the basis of the solution space W [itex]\subset[/itex] [itex]\Re^{4}[/itex]
of the system of linear equations

[itex]2x_{1}[/itex] + [itex]1x_{2}[/itex] + [itex]2x_{3}[/itex] +[itex]3x_{4}[/itex] =0
[itex] _{ }[/itex]
[itex]1x_{1}[/itex] + [itex]1x_{2}[/itex] + [itex]3x_{3}[/itex] = 0


Homework Equations


The basis must span W and be independent.


The Attempt at a Solution


Solving the above system, I get
[itex]x_{2}[/itex] = [itex]-x_{1}[/itex] - [itex]x_{3}[/itex]
[itex]x_{4}[/itex] = [itex]\frac{x_{3}-x_{1}}{3}[/itex]

With 2 degrees of freedom, [itex]x_{1}[/itex] and [itex]x_{3}[/itex],
so I must need a 2D basis. I separately fixed [itex]x_{1}[/itex] and [itex]x_{3}[/itex] to 1 and the other to zero and got the following vectors:
[1, -1, 0, -1/3] and [0, -3, 1, 1/3]
I feel like this is right, as I've been looking up some examples, but I'm not sure this spans all the solutions.

Am I on the right track?

Yes, your vectors span W.

A more systematic way to do things is the row-reduce your matrix, which gives this matrix:
[tex]\begin{bmatrix}1&0&-1&3\\0&1&4&-3 \end{bmatrix}[/tex]

From this matrix you can read off your solutions as
x1 = x3 - 3x4
x2 = -4x3 + 3x4
x3 = x3
x4 = ... x4

From this you might be able to see that any vector x in the solution space is a linear combination of these two vectors: <1, -4, 1, 0>T and <-3, 3, 0, 1>T.
 
  • #3
To add just a little bit, you are saying that for any vector in the solution space, [itex]<x_1, x_2, x_3, x_4>[/itex], we must have [itex]x_2= -x_1- x_3[/itex] and [itex]x_4= (1/3)x_3- (1/3)x_1[/itex]. That is, [itex]<x_1, x_2, x_3, x_4>= <x_1, -x_1- x_3, x_3, (1/3)x_3- (1/3)x_4>= x_1<1, -1, 0, -1/3>+ x_3<0, -1, 1, 1/3>[/itex] which makes it clear what a basis is.

Mark44 is saying that [itex]x_1= x_3- 3x_4[/itex] and [itex]x_2= -4x_3+ 3x_4[/itex] so that [itex]<x_1, x_2, x_3, x_4>= <x_3- 3x_4, -4x_3+ 3x_4, x_3, x_4>= x_3< 1, -4, 1, 0>+ x_4<-3, 3, 0, 1>[/itex]. That gives another basis for the same subspace.
 

1. What is algebra?

Algebra is a branch of mathematics that deals with the manipulation of symbols and solving equations using rules and operations such as addition, subtraction, multiplication, and division.

2. How is algebra used in real life?

Algebra is used in many areas of daily life, such as calculating budgets, understanding patterns and relationships, and solving everyday problems. It is also essential in fields such as engineering, physics, and economics.

3. What is a solution space in algebra?

A solution space in algebra is a set of all possible solutions to a given equation or system of equations. It represents the values of the variables that make the equation(s) true.

4. How is algebra related to other branches of mathematics?

Algebra is closely related to other branches of mathematics, such as geometry, calculus, and number theory. It provides the foundation for these subjects and is used to model and solve problems in these areas.

5. Why is algebra important?

Algebra is important because it helps develop critical thinking and problem-solving skills. It also provides a powerful tool for modeling and solving real-world problems, making it an essential skill for many careers and academic fields.

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