Answer:
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.
