Subsets, Projection Maps, Basis and Direct Sums
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Let n >= 1. Define the subsets U and W in V = F^n as follows:
U = {(x_1, . . . , x_n) : x_1 + . . . + x_n = 0} W = {(x_1, . . . , x_n) : x_1 = . . . = x_n}
a) Prove that U and V are subspaces of V .
b) Prove that V = is the "direct sum" of U and W.
c) Let (v_1, . . . v_n) = ((1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, . . . , 0, 1)) be the standard basis of V, and let
P_U,W and P_W,U denote the projection maps. Find P_U,W(vi) and P_W,U(vi).
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Solution Summary
Subsets, Projection Maps, Basis and Direct Sums are investigated. The solution is detailed and well presented.
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