Does W Qualify as a Subspace of R^3?

[Robb]

Determine whether or not W is a subspace of R^3, where W consists of all vectors (a, b, c) in R^3 such that (a) a = 3b; (b) a<=b<=c; (c) ab = 0.

a) Because vectors (a,b,c) can assume any value in W, W is a subset of R^3. Also, the zero vector belongs to W and W is closed under vector addition (a,b,c are elements of W, a+b+c belongs to W) and scalar multiplication (a,b,c belong to W, k belongs to K and k(a,b,c) belongs to W.

Please give input on my statement (a).

 

[andrewkirk]

W is closed under vector addition (a,b,c are elements of W, a+b+c belongs to W)

That is not what the vector addition axiom is about. a,b,c are components, not vectors. The vector addition axiom requires that, if (a,b,c) and (d,e,f) are both in W, then their sum, which is (a+d, b+e, c+f) is also in W.

a,b,c belong to W, k belongs to K and k(a,b,c) belongs to W

Are you sure? What if k is negative?

 

[Robb]

That is not what the vector addition axiom is about. a,b,c are components, not vectors. The vector addition axiom requires that, if (a,b,c) and (d,e,f) are both in W, then their sum, which is (a+d, b+e, c+f) is also in W.
Are you sure? What if k is negative?

I guess I assumed "where W consists of all vectors (a,b,c)" is saying they are vectors. So, may I make assumptions with this problem as you did above with (a,b,c) and (c,d,e)? If so, what does that tell me about (a) a=3b?

 

[Mark44]

I guess I assumed "where W consists of all vectors (a,b,c)" is saying they are vectors.

There is more to that sentence - "such that (a) a = 3b; (b) a<=b<=c; (c) ab = 0."

Take two arbitrary vectors in W. Add them. Is their sum also in W? I.e., does this sum satisfy the three conditions above.
Take an arbitrary vector in W and an arbitrary constant. Multiply them. Is this new vector also in W? Again, it has to satisfy the same three conditions.

These are what you have to do to show that W is a subspace of R³. If either one or both of the above fail, W is not a subspace of R³.

 

[Robb]

There is more to that sentence - "such that (a) a = 3b; (b) a<=b<=c; (c) ab = 0."

Take two arbitrary vectors in W. Add them. Is their sum also in W? I.e., does this sum satisfy the three conditions above.
Take an arbitrary vector in W and an arbitrary constant. Multiply them. Is this new vector also in W? Again, it has to satisfy the same three conditions.

These are what you have to do to show that W is a subspace of R³. If either one or both of the above fail, W is not a subspace of R³.

I thought these were three independent questions as the book gives three independent answers. I was looking at each individually.

 

[Mark44]

I thought these were three independent questions as the book gives three independent answers. I was looking at each individually.

OK, I misunderstood what the question was.
For (a), W = {(a, b, c) | a = 3b}
Take two arbitrary vectors u and v in W. Add them. Is their sum u + v also in W?
Take an arbitrary vector u in W and a scalar k. Is ku in W?
Do the same for the other two parts of the question.