# Subsets and subspaces of vector spaces

1. Nov 10, 2009

### gtfitzpatrick

1. The problem statement, all variables and given/known data

T = {(1,1,1),(0,0,1)} is a subset of R$$^{3}$$ but not a subspace

sol

i have to prove it holds for addition and scalar multiplication

so let x=(1,1,1) and y =(0,0,1) so x+y = (1,1,2)

so it holds

let $$\alpha$$ = a scalar
then $$\alpha$$x = ($$\alpha$$,$$\alpha$$,$$\alpha$$)
and $$\alpha$$y = (0,0,$$\alpha$$)

so that holds.

i think i've shown that it is a subpace but the question says it isnt?

Last edited: Nov 10, 2009
2. Nov 10, 2009

### Staff: Mentor

Re: suspace

I don't think you have given us the exact wording of this problem. The two vectors you gave are a basis for and span a two-dimension subspace of R^3.

3. Nov 10, 2009

### Staff: Mentor

Re: suspace

Thinking about this some more, you have a set T with two vectors in it. With x and y as before, is x + y in the set? Is cx in the set for an arbitrary scalar?

4. Nov 10, 2009

### gtfitzpatrick

Re: suspace

i think they are in the set, x+y = (1,1,2) which is in R^3 and cx= (c,c,c) which is also in R^3 for any scalar c

5. Nov 10, 2009

### Staff: Mentor

Re: suspace

The set T is {(1, 1, 1), (0, 0, 1)}. How can you say that x + y is in this set? If c = 2, is c(1, 1, 1) in this set?

6. Nov 10, 2009

### gtfitzpatrick

Re: subspace

am i not just to show that they are in R^3?

7. Nov 10, 2009

### Staff: Mentor

Re: suspace

Do you know the definition of a subspace of a vector space?