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.
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!
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.
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)
If you need to show that some space and two operations are a vector space, yes, you have to verify all 10 axioms. 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.
Correction to your terminology: (1, 1) and (1, -1) are elements in the subset of R^{2}. You have shown that this subset is not a subspace of R^{2}, so you shouldn't call it a subspace.