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April 29th, 2011, 09:56 PM   #1
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Proving something is a subspace by proving it is a span.

I am asked to prove that something is a vector subspace and then find the basis of it. I know that going through all the axioms is pretty time consuming and I think the teacher is trying to get us to do it another way. I think that way is proving that it is a span. So here's what I've been given:

I can't get latex to work with me right now so I'm just going to write it out like this

where a,b,c are real numbers.

I have to prove that H is a vector subspace and find a basis for H
Then I have to determine whether the basis for H spans the set R^4.

I have been looking through my books for about 2 hours now and this is my last resort. I have tried, but just can't figure it out. Help is appreciated.
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April 29th, 2011, 10:57 PM   #2
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Re: Proving something is a subspace by proving it is a span.


To show that a subset of a vector space is a subspace, you need to show that the subset is closed under vector addition and scalar multiplication. So, if you are presented with a subset H ? V, you will want to take two arbitrary members of this set, and see whether their sum is in the set, and whether either of the members, when multiplied by an arbitrary scalar, is also in the set.

To find a basis, you need a linearly independent spanning set. The first step is to make sure that your set is not already a basis. To do this, you write a linear combination of all the members of H and show that the zero vector is never a non-trivial such combination.

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