Question

a) Find a basis for W2 = {(x, y, z) ∈ R 3 : x + y + z = 0} (over R). Justify your answer. What is the dimension of W2?

b) Find a basis for W4 = {(x, y, z) ∈ R 3 : x = z, y = 0} (over R). Justify your answer. What is the dimension of W4?

Answer 1

a)

Observe that (1,0,-1) and (0,1,-1) are in W₂ as 1+0+(-1)=0 and 0+1+(-1)=0.

Let us prove that (1,0,-1) and (0,1,-1) are linearly independent over |R. Suppose that a(1,0,-1) + b(0,1,-1) = (0,0,0) for some reals a,b. This implies that (a,b,-a-b)=(0,0,0). Thus, a=0 and b=0. Thus, (1,0,-1) and (0,1,-1) are linearly independent over |R.

Now suppose that (x,y,z) is in W₂. Then, x+y+z=0 and hence, z=-x-y. Observe that (x,y,z) = (x,y,-x-y) = x(1,0,-1) + y(0,1,-1). Hence, span{ (1,0,-1),(0,1,-1) } = W₂.

Hence, { (1,0,-1),(0,1,-1) } is a basis of W₂ over |R. Thus, dimension of W₂ is 2.

b)

Observe that (1,0,1) is in W₄ = { (x,y,z) |R³ : x = z, y = 0 }. Now, { (1,0,1) } is linearly independent over |R.

Further, let (a,b,c) be in W₄. By definition of W₄, a=c and b=0. Thus, (a,b,c)=(a,0,a)= a(1,0,1). Thus, span{ (1,0,1) } = W₄.

Thus, { (1,0,1) } is a basis of W₄. And dimension of W₄ = 1.