Homework SourseDetermine which of the following subsets of R3 are subspaces
Determine which of the following subsets of R^3 are subspaces. Be sure to justify your answers. W = {(x, y, z): z greaterthanorequalto 0}. U = {(x, y, z): x + y + z= 0}
Solution
(a) Let p be an arbitrary scalar and let (x,y,z) be an arbitrary element of W. Then p(x,y,z) = (px,py,pz). Now, if p is negative, then pz 0 . Thus p(x,y,z) W if p is negative. Thus, W is not closed under scalar multiplication, and hence W is not a subspace of R3.
(b) Let p be an arbitrary scalar and let (x1,y1,z1) and (x2,y2,z2) be 2 arbitrary elementsof U.Then (x1,y1,z1)+(x2,y2,z2) = ( x1 +x2 , y1+y2, z1+z2) . Now, x1 +x2 + y1+y2 + z1+z2 = (x1+y1+z1) + (x2+y2+z2) = 0+0 = 0. Therefore, (x1,y1,z1)+(x2,y2,z2) U. Further, p(x1,y1,z1) = (px1, py1,pz1). Since px1+ py1+pz1= p(x1+y1+z1) = p*0 = 0, hence p(x1,y1,z1) U. Therefore U is also closed under scalar multiplication. Hence U is a subspace of R3.
