Let V=R2 and let H be the subset of V of all points on the line 4x+3y=12. Is H a subspace of the vector space V?

First, we define:

V = R^2.

This means that V is the set of all the points (x, y)

Such that:

x ∈ R

y ∈ R

Then:

(x, y) ∈ R^2.

Ok, now we define H as:

The set of all points in the line 4x + 3*y = 12

we first can write this equation as:

3y = 12 - 4x

y = 4 - (4/3)*x

Then we can write all the elements in H as:

(x, 4 - (4/3)*x)

Now the question:

H is a subspace of V if:

1) for v and u ∈ H, then u + v ∈ H

2) for v ∈ H, then a*v ∈ H, such that a ∈ R.

Let's see if those things are true.

1) v = (v, 4 - (4/3)*v)

  u = (u, 4 - (4/3)*u)

then we must have that:

v + u = (v + u, 4 - (4/3)*(v + u)) ∈ H

let's check:

v + u = (v, 4 - (4/3)*v) + (u, 4 - (4/3)*u) = (v + u, 8 - (4/3)*(u + v))

This is different than the thing we wanted to find, because:

(v + u, 8 - (4/3)*(u + v)) ∉ H

So the sum of two elements of H is not an element of H, this means that H is not a subspace.