Verify that W is a subspace of V Assume that V has the stand

Verify that W is a subspace of V. Assume that V has the standard operations

{(x,y,4x-5y): x and y are real numbers} V=R^3

Solution

Is Zero a vector in W.

At x = 0, y = 0, 4x-5y = 0 So (0,0,0) is vector in both V and W.

(b)Let (x1, y1, 4x1 -5y1) and (x2 , y2, 4x2 -5y2) be two elements.

Is the subspace closed under addition.

So u+v = (x1+x2 , y1+y2 , 4(x1+x2)-5(y1+y2))

So it is closed under addition.

Similarly is it closed under multiplication.

Consider C a scalar.

c* {(x,y,4x-5y)} = {(cx,cy, 4cx - 5cy)} .

Closed under multiplication.

So W is a subspace of V.

Hence proved