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## Homework Statement

The question asks to show whether the following are sub-spaces of R^3. Here is the first problem. I want to make sure I'm on the right track.

Problem: Show that W = {(x,y,z) : x,y,z ∈ ℝ; x = y + z} is a subspace of R^3.

## Homework Equations

None

## The Attempt at a Solution

Okay, so here is my attempt:

Since y,z ∈ ℝ

y+z ∈ ℝ because ℝ is closed under addition.

So x ∈ ℝ.

:Break:

Let u = (x,y,z) be some vector in W.

Let v = (a,b,c) be some vector in W.

(x,y,z) + (a,b,c) = (x+a, y+b, z+c)

x+a ∈ ℝ because ℝ is closed under addition and x,a ∈ ℝ by the previous logic.

y+b ∈ ℝ because ℝ is closed under addition and y,b ∈ ℝ.

z+c ∈ ℝ because ℝ is closed under addition and z,c ∈ ℝ.

x and a are still dependent on y,z and b,c respectively.

So u + v ∈ W

:Break:

Let k be some scalar on ℝ

Let (x,y,z) be some vector u on ℝ.

k(x,y,z) = (kx,ky,kz)

Since ℝ is closed under multiplication and x,y,z ∈ ℝ (as already demonstrated), kx ∈ ℝ, ky ∈ ℝ, kz ∈ ℝ

kx = ky + kz, so kx remains dependent on ky, kz.

So ku ∈ W

:Break:

So u is a subspace of ℝ.

Am I missing anything? All the examples I've found are extremely poor and dissimilar to the specified problem.

Also, since W has 3 real valued components, isn't W technically defined on ℝ^3, not ℝ^2? How can it be a subspace if this is the case?