Is W a subspace of V?

Question

Let V=R^3 and let W={(x, y, z) ∈ R^3| x=3y}

Is W a subspace of V?

If it is, find a basis and dimension for this subspace

Answer 1

To show that W is a subspace it's enough to show that W is closed under addition and is also closed under scalar multiplication.

Let (a,b,c) and (d,e,f) be elements of W. So, a = 3b and d = 3e. Then, (a, b, c) + (d, e, f) = (a+d, b+e, c+f). 3(b+e) = 3b + 3e = a + d. So, since the x-coordinate is equal to 3 times the y-coordinate, (a,b,c) + (d,e,f) is in W. Therefore, W is closed under addition.

Let (x,y,z) be an element of W and let c be a scalar (i.e., a real number.). Then x = 3y.

So, c(x, y, z) = (cx, cy, cz). 3(cy) = c(3y) = cx. So, since the x-coordinate of c(x,y,z) is 3 times the y-coordinate, c(x,y,z) is in W. Thus, W is closed under scalar multiplication.

So, W is a subspace of llR³.

Any element of W can be written as (3y, y, z) = y(3, 1, 0) + z(0, 0 1). This is a plane in lR³, which is 2-dimensional.

So, a possible basis for W is {(3, 1, 0), (0, 0, 1)}