Subsets and subspaces of vector spaces

[gtfitzpatrick]

Homework Statement

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 \alphax = (\alpha,\alpha,\alpha)
and \alphay = (0,0,\alpha)

so that holds.

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

 

[Mark44]

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.

 

[Mark44]

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?

 

[gtfitzpatrick]

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

 

[Mark44]

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

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?

 

[gtfitzpatrick]

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

 

[Mark44]

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