Subspace Questions: Checking 2 Sets in R^3

[EvLer]

I am not sure about these 2 whether they are subspaces or not (i do know how to check whether it is a subspace or not)

subsets of R^3, subspace or not?
1.all combinations of (1,1,0) and (2,0,1)
2.plane of vectors (b1,b2,b3) that satisfy b3-b2+3b1 = 0

thanks for help.

 

[StatusX]

What does a subspace mean to you?

 

[EvLer]

closed under addition and scalar multiplication... add two vectors and they should still be in the space, multiply by a const and the vector is still in space... also has to contain zero.
i am not sure which side to approach THESE two...

and what in the world is meant by "all combinations"?

 

[StatusX]

By combinations, I assume they mean linear combinations. A linear combination of a set of vectors v₁,...,v_n is any vector of the form a₁ v₁+...+a_n v_n where the a_k are scalars. The set of such vectors is clearly a subspace, and this is sometimes taken as the defintion. For the other one, assume two vectors satisfy the equation and show their sum and scalar multiples do as well.

 

[EvLer]

For the other one, assume two vectors satisfy the equation and show their sum and scalar multiples do as well.

scalar multiple I get, but for sum not exactly sure:
(u3+v3) - (u2+v2) + 3(u1+v1) = 0
what does that give me?

 

[StatusX]

u3-u2+3u1=0, and similarly for v, so...

 

[EvLer]

oh ok, thanks.

one other quick question: if I factor a constant out of a row of a matrix B and get matrix A then can i say that B = 2A?
what if i factor a constant out of a column?
thanks again...

edit: also, how would i show that column spaces of 2 matrices are equal?