Lin Alg - Basis of the Space of Solutions of a Linear System

In summary, the book claims that for part (c) the dimension is 1, but that basis is not a solution to both equations. For part (b) the elements in the book's basis is a solution to the system, but for part (d) there is no basis.
  • #1
mattmns
1,128
6
Just a quick question. If I have a system of linear equations, and I find a basis for the space of solutions, then each element in the basis should be a solution to the system right? Thanks!
 
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  • #2
What do you think?
 
  • #3
I think yes, but if so, then my book has a wrong answer.
 
  • #4
Well, you should be right. Maybe it means that the system of equations could be nonhomogenous and the "space of solutions" they are talking about is only for the related homogenous system. Or maybe they are just wrong.
 
  • #5
I am dealing with homogenous systems, sorry for not specifying that earlier. The book has 4 problems like this, 3 of which have a bases, 2 of which are not solutions to the entire solution. Here is the actual problem and answer from the book.

----------------

What is the dimension of the space of solutions of the following system of linear equations? In each case, find a basis for the space of solutions.
(a)
2x + y - z = 0
2x + y + z = 0
-----
The dimension is 1, and the books basis is (1,-1,0)
-----
But this element is not a solution to the above system. In my opinion, the basis should be (1, -2, 0). [fixed typo]

then for part (c)
[itex]4x + 7y - \pi z = 0[/itex]
[itex]2x - y + z = 0[/itex]
------
The book then says that the dimension is 1, which is what I got, but it says that the basis is [tex] ( \frac{\pi - 3}{10} , \frac{ \pi + 2}{5} , 1 ) [/tex]

But this basis is not a solution to both equations (it does not work for the first equation), and therefore not a solution to the system.
-------------

And, for part (b) the elements in the book's basis is a solution to the system, and for part (d) there is no basis.

Am I missing what the book is getting at (there are no examples of a similar problem), or do you think the book is just wrong? Thanks.
 
Last edited:
  • #6
Well, for a the correct answer is (1, -2, 0). For c the dimension is actually 2.
 
  • #7
0rthodontist said:
Well, for a the correct answer is (1, -2, 0). For c the dimension is actually 2.
You are right about a, that was my typo :smile:

But for c how are you getting the dimension as 2?
 
  • #8
Ah, my mistake, never mind.

I don't know why your book is having so many errors. Maybe you are looking at the wrong solutions?
 
  • #9
lol, no they are the right "solutions."

Thanks for the help!
 

1. What is the basis of the space of solutions of a linear system?

The basis of the space of solutions of a linear system is a set of vectors that can be used to represent all possible solutions to the system. It is a fundamental concept in linear algebra and is used to understand the structure of the solution space.

2. How do you find the basis of the space of solutions of a linear system?

The basis of the space of solutions can be found by first solving the linear system using techniques such as Gaussian elimination or matrix inversion. Then, the columns of the resulting solution matrix that contain the leading variables form the basis of the solution space.

3. Can the basis of the space of solutions be unique?

Yes, the basis of the space of solutions can be unique. This occurs when the linear system has a unique solution or when the solution space is one-dimensional, meaning that there is only one linearly independent vector that can represent all solutions.

4. How does the dimension of the basis of the space of solutions relate to the number of variables in the linear system?

The dimension of the basis of the space of solutions is equal to the number of variables in the linear system minus the rank of the coefficient matrix. This means that the dimension of the basis can vary depending on the number of variables and the structure of the linear system.

5. Can the basis of the space of solutions change if the linear system is transformed?

Yes, the basis of the space of solutions can change if the linear system is transformed. This can happen if the transformation affects the rank of the coefficient matrix or if the transformation introduces new variables to the system, changing the dimension of the solution space.

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