Subspace questions

1. Sep 16, 2006

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.

2. Sep 16, 2006

StatusX

What does a subspace mean to you?

3. Sep 16, 2006

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"?

4. Sep 16, 2006

StatusX

By combinations, I assume they mean linear combinations. A linear combination of a set of vectors v1,...,vn is any vector of the form a1 v1+...+an vn where the ak 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.

5. Sep 16, 2006

EvLer

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

6. Sep 16, 2006

StatusX

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

7. Sep 16, 2006

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?

Last edited: Sep 16, 2006