Consider the subset W = {(a, b, c, d) | 2b - c + 2d = 0} R4. Show that W is a su
ID: 2943255 • Letter: C
Question
Consider the subset W = {(a, b, c, d) | 2b - c + 2d = 0} R4. Show that W is a subspace of R4. Find a basis BW for W and state dim(W). Describe W geometrically. Show that the set S = {(1, 1, 0, -1)T, (-1, 0, 2, 1)T, (1, 2, 2, -1)T, (1, 1, 2, 0)T, (0, 1, 0, -1)T} is a spanning set for W but not a basis. Find another basis, BS, for W that consists of a selection of the vectors in S. Determine the coordinate vectors of the dependent vectors in S with respect to the ordered basis BS found in (v). Would the set BW {(0, 2, -1, 2)T} be a basis for R4? Explain your answer.Explanation / Answer
1. Clearly W is non empty since (0,0,0,0) is in W.
Let (a,b.c,d),(e,f,g,h) be in W. Then
2b-c+2d = 0 and 2f-g +2h = 0________________(*)
Consider (a,b.c,d)+ (e,f,g,h) = (a+e, b+f,c+g, d +h) is in W since
2(b+f) - (c+g) + 2 (d+h) = 2b+2f - c -g +2d +2h = 0 ( by (*))
Let α be in R and (a,b.c,d) in W .Then α(a,b,c,d) = (αa,αb,αc,αd) is in W since α(2b-c+2d ) = 0
Thus W is a subspace of R4