Is W a subspace of V?

• Jan 16th 2013, 11:49 PM
TimsBobby2
Is W a subspace of V?
Each of the following involves a vector space V and a subset W. For each decide whether W is a subspace of V.

1.) V = R3, W={(x,y,z) | x <= z } I say no because it doesn't preserve scalar multiplication. For example, if you have (2,2,4) 2 is <=4 but -2 >= -4 which contradicts that x must be less than or equal to z. Am I right?

2.) V = R<= 3 [x], W = Z<= 3 [x]. So V is the polynomials in x with real coefficients and degree at most 3. W is the polynomials in x with integer coefficients and degree at most 3. I'm not sure on this one, any help?
• Jan 17th 2013, 12:19 AM
chiro
Re: Is W a subspace of V?
Hey TimsBobby2.

Can you show us an attempt to prove your statements with the appropriate vector space axiom?
• Jan 17th 2013, 12:22 AM
Deveno
Re: Is W a subspace of V?
1) yes, multiplying by a negative scalar ruins everything.

2) is it closed under polynomial addition? is it closed under scalar multiplication? (recall that (cp)(x) = c(p(x)) for all x). is the 0-polynomial in W?

you might want to ask yourself: for the polynomial p(x) = x in Z[x], is (1/2)p(x) in Z[x]?
• Jan 17th 2013, 12:35 AM
TimsBobby2
Re: Is W a subspace of V?
So for 2) if I consider W and I do scalar multiplication by 1/2, (1/2)p(x) is not in Z[x], so W is not a subspace of V? Is that correct to say (I'm just starting this stuff today)
• Jan 17th 2013, 01:53 AM
Deveno
Re: Is W a subspace of V?
what do you think? i'm not trying to be mean....i'm asking you to convince yourself that what you say is true. the truth of math doesn't depend on "who the expert is", it should be self-evident. if you are unsure about something, ask about that. understanding the ideas is the important part....getting the answers correct is only a useful by-product.