Linear Algebra help :(

iPancakes x

Hi,

I have a homework assignment for Linear Algebra and I have done most of it right so far but I really cannot understand these two questions

Q1. Which of the following subsets of P2 are subspaces of P2?

A. { p(t) | p'(t) + 7 p(t) + 9 =0}
B. { p(t) | p(-t)= p(t) for all t}
C. { p(t) | p'(t) is constant}
D. { p(t) | int from 0 to 3 p(t)dt = 0}
E. { p(t) |p'(2)= p(8)}
F. { p(t) | p(6)= 4}

Q2. Which of the following sets are subspaces of R^3?

A. { (x,y,z) | x + y + z = 6}
B. { (x, x+2, x - 8) | x arbitrary number}
C. { (-6 x - 7 y, 4 x - 9 y, 3 x + 9 y ) | x,y arbitrary numbers}
D. { (x, 0, 0) | x arbitrary number}
E. { (x,y,z) | x < y < z}
F. { (x,y,z) | x + y + z = 0}

I'd really appreciate any help anybody could give me!!

Thanks!

sigmundv

I assume they mean "linear subspace". Then you need to apply the following theorem in each case; if it holds, the subset is a subspace.

Theorem:
Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if W satisfies the following three conditions:
1) The zero vector, 0, is in W.
2) If u and v are elements of W, then the sum u + v is an element of W;
3) If u is an element of W and c is a scalar from K, then the product cu is an element of W.

http://en.wikipedia.org/wiki/Linear_subspace