Question

Which one of these are subspaces?

Answer 1

\[V _{1}=\left\{ {(x_{1}, x_{2}, x_{3}) : x_{1}+x_{2}+x_{3} \ge 0} \right\}\]
\[V _{2}=\left\{ {(x, y, z) : x-2y+4z=1} \right\}\]
\[V _{3}=\left\{ {(x, y, z) : x^{2}+4z=0} \right\}\]
\[V _{4}=\left\{ {(x, y, z) \in \mathbb{R}^{3} : xyz \ge0} \right\}\]
\[V _{5}=\left\{ {(x, y, z) \in \mathbb{R}^{3} : x+y=0} \right\}\]
\[V _{6}=\left\{ {(l, m, n) \in \mathbb{R}^{3} | l,m,n \in Z} \right\}\]

Answer 2

V2 are not subspace because we dont have 0 vector

Answer 3

v5 are subspace.because its satisfies the three conditions

Answer 4

I guess we are talking about vector spaces?

Answer 5

Yes

Answer 6

so what are the rules for a subspace?

Answer 7

This confuses me a bit to be honest.
I think you can only refer to a subspace with respect to another vector space.

Answer 8

Which one of these are subspaces? of what? eachother? Real numbers?

Answer 9

im guessing V_1 is a subspace of R^3

Answer 10

I mean if it is a subspace then its a subspace of R^3

Answer 11

In that case all of them are subspaces apart from number 2

Answer 12

it actually does not matter, as long as we know that they are subsets of some vector space.

Answer 13

OK that is true, I understand that

Answer 14

so let a be in R, let b,c be in V_1
you need to show that a(b+c) = is in V_1

Answer 15

@dw3 does it say what the space is?

Answer 16

I was a bit too hasty, forgot about -scalar product. That will rule out a few more

Answer 17

I think 3,5,6

Answer 18

No, it does say what space :/ The question was written as I posted

Answer 19

v6 are not subspace .scalar product not satisfy

Answer 20

I'm pretty sure it has to be a subspace of another space… Else, its just a space.

Answer 21

Thanks for input everyone. I was thinking V1 was a subspace, but after reviewing class material, I had realized that it did not satisfy the scalar multiplication. Only 5 was a subspace. Thanks!