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Subspace Questionsby Mooey
Tags: subspace 
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#1
Mar910, 06:00 PM

P: 3

1. The problem statement, all variables and given/known data
Determine whether the following sets form subspaces of R2: { (X1, X2)  X1 = X2 } { (X1, X2)  (X1)^2 = (X2)^2 } 2. Relevant equations 3. The attempt at a solution I'm clueless. I've been trying to figure it out for a good thirty minutes on both of them, but I'm completely stuck. 


#2
Mar910, 06:02 PM

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P: 21,280

How can you tell if some subset of a vector space is a subspace of that vector space?



#3
Mar910, 06:47 PM

P: 3

You check to see if there's a counterexample and/or if it's closed under addition/multiplication. Problem is my book has given no examples of something like these, and my professor went over it the day I was out of class so I have no idea how to work it out.
Edit: Okay, I've figured out the first. I took (1, 1) and (1, 1) which are a part of the subspace, but their sum is not (i.e. (2,0) fails since x1 =/= x2)). I could still use help on the second though! 


#4
Mar910, 07:05 PM

P: 234

Subspace Questions
There is a list of axioms, the vector space axioms, somewhere in your book. A subset of a vector space is a subspace iff it is a vector space in its own right, under the same operations.
So all you need to do is check the axioms. 


#5
Mar910, 07:15 PM

P: 3

Nevermind, I used my above example (1, 1) and (1,1) for the second question and figured it out!
(1, 1) is a part of the subspace, and so is (1, 1), but their sum is not (2,0) (i.e. 2^2 =/= 0^2) 


#6
Mar910, 07:28 PM

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OTOH, if you are given a subset of a vector space (R^{2} in the OP's problem), all you need to do is check that 0 is in the subset, and that the subset is closed under vector addition and scalar multiplication. 


#7
Mar910, 07:30 PM

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P: 21,280




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