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

[mattmns]

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!

 

[0rthodontist]

What do you think?

 

[mattmns]

I think yes, but if so, then my book has a wrong answer.

 

[0rthodontist]

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.

 

[mattmns]

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.

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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
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The dimension is 1, and the books basis is (1,-1,0)
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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)
$4x + 7y - \pi z = 0$
$2x - y + z = 0$
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The book then says that the dimension is 1, which is what I got, but it says that the basis is $$( \frac{\pi - 3}{10} , \frac{ \pi + 2}{5} , 1 )$$

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.
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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.

 

[0rthodontist]

Well, for a the correct answer is (1, -2, 0). For c the dimension is actually 2.

 

[mattmns]

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

But for c how are you getting the dimension as 2?

 

[0rthodontist]

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?

 

[mattmns]

lol, no they are the right "solutions."

Thanks for the help!