Subspace Questions: Determine Sets in R2

[Mooey]

Homework Statement

Determine whether the following sets form subspaces of R2:

{ (X1, X2) | |X1| = |X2| }
{ (X1, X2) | (X1)^2 = (X2)^2 }

Homework Equations

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.

 

[Mark44]

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

 

[Mooey]

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!

 

[Tinyboss]

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.

 

[Mooey]

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)

 

[Mark44]

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.

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 (R² 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.

 

[Mark44]

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!

Correction to your terminology: (1, 1) and (1, -1) are elements in the subset of R². You have shown that this subset is not a subspace of R², so you shouldn't call it a subspace.