Thread: linear algebra question: Subspaces

1. linear algebra question: Subspaces

Need a little help figuring whether the following subsets are actually subspaces:

1. the plane of vectors (b1, b2, b3) with b1=b2
2. the plane of vectors with b1 = 1
3. all linear combinations of v=(1,4,0) and w=(2,2,2)
4. all vectors that satisfy b1+b2+b3 = 0
5. all vectors with b1<= b2 <= b3

i'm having difficulties figuring which one satisfy the subspace requirements and how exactly the prove it... please help!!!

2. A set is a subspace if and only if all of the following conditions are satisfied:

It is contained in or equal to a vector space.
It is non-empty
It is closed under multiplication by a scalar
It is closed under addition.

The first 2 of these are generally very easy to prove. For the 3rd use the fact that it is non-empty and assume that u is in the set and prove that this implies ku is in the set for any scalar k. For the third, assume that u and v are in the set and show that this implies u+v is in the set.