Question

Consider the following subspaces of R3: U = {(x1, 22, 23) : x1 = x2 + x3}, V = {(21, X2, X3) : x1 = x2}, W = {(21, 22, 23) : x1 = 22 = x3}.

Please help provide answers with explanation (emphasizing on explanation). Will rate for prompt, correct, and complete answers. Thank you very much.

Answer 1

We will find basis for U,V and k let U C,0), (,011) Basis USpan C.0), 1,01) where =12 x (1,2, *3) Let (2,2 3) Basis for V N Span (l0), (0,011) where X = 2 l3 хE k Let 3C 1 Basis for = {C)} Span C) Span 0) C01 S pan o) Co,0,) SpanC and W cR 3. f4o10) (HLO)+(+ehort)t 1)

(,1,0)4(011) Span (110), (,0,1)} = SpantC) - (,0,0) E Ut (oo) o (,,o) ) (1011) E Span{(ll,0), C101 U Coio,) 1) (10) o (oit)C) (o101)E U + \J :. (1,o,0) , (o,10) (o,011) is n Ufk CI010), Co,0), (o,0,) U 010), (o,0), (o,0 1) Ut Span U s a subspace) To show V+ J = V. C veV V= v+ Co,0,0) V+ k v€ V+_ V SV will show Se Now e V Let 1

ve V, wek where X= 12 2= Y3 where ** (x1, A2 1 %3) + (9, ,92,) = (xi+, 2+92, 3+) XV by , ** IR and 3 1 2+Y2 C , 12492, %3+Y ohere xit, = x2+ 92 put Z3 3 Z Z2, Za) Z1=Z2 e V+ then V V