Determining whether or not (a,0,0) and (a,b,0) are subspaces of R3

[tnorman]

Homework Statement

"use theorem (below) to determine which of the following are subspaces of R3:

(a,0,0) and (a,b,0)

Homework Equations

The theorem: W is a subspace of V iff:

- u and v are vectors in W, u + v is in W
- k is a scalar, u is a vector in W, then ku is in W

The Attempt at a Solution

I thought that I understood that (a,0,0) was a subspace of R3 since if we take (a,0,0) and (b,0,0) we get (a+b,0,0) which is still in the subspace, the solution says that this is a subspace of R3 but then I thought the same idea would hold for (a,b,0) since if we take (a,b,0) and (c,d,0) we would get (a+b,c+d,0) but the solution sheet says it is not, however my solution sheet doesn't elaborate on why, its just a yes/no answer.

(PS I also thought multiplying by a scalar would hold too)

Can anyone help me out? I having a heck of a time truly understanding vector spaces. Thanks in advance

 

[Coto]

You are correct. The set of all (a,b,0) is a vector subspace of R3. In fact, it is exactly the xy-plane R2 as visualized in R3.

Likewise the set of all (a,0,0) is simply the x-axis as visualized in R3.

It's actually a much easier concept that you'd expect. I remember struggling on it somewhat too, but if you just follow the definitions of what a vector subspace or vector space is, and stick to them when you're trying to show something is a vect. space, you'll be fine (as you were above.)

 

[HallsofIvy]

My first reaction was "No, (a, 0, 0) is not a subspace of R³, it is a vector in R³", but what you really mean is:

Is V= {(a, 0, 0)| a a real number}, with the same definitions of addition and scalar multiplication as in R³, a subspace of R³?

Use the definition of "subspace"! It is clearly a subset of R³ but does it satify the conditions for a vector space? Fortunately most of the conditions are automatically satisfied because because addition and scalar multiplication already satisfy the necessary condition in R³. All you really need to do is show that the set is closed under those operations. If a and b are real numbers then (a, 0, 0) and (b, 0, 0) are in V. Is (a, 0, 0)+ (b, 0, 0) in V? That is, is it of the form (c, 0, 0) for some real number c? Similarly, if a and x are real numbers so that (a, 0, 0) is in V, is the scalar product, x(a, 0, 0) in V?

 

[FabricWarp]

Right but he doesn't have a problem with (a,0,0), he already knows and proved it as he stated above, what he wants to know is how is it that vectors of the form (a,b,0) are not a subspace? because the ACTUAL Answer in the TEXTBOOK IS it ISN'T a subspace.

 

[Coto]

Right but he doesn't have a problem with (a,0,0), he already knows and proved it as he stated above, what he wants to know is how is it that vectors of the form (a,b,0) are not a subspace? because the ACTUAL Answer in the TEXTBOOK IS it ISN'T a subspace.

His book is wrong.