is the given subset of W of R^3 a subspace? Solution Yes, Proof ---------- a) W

Question

is the given subset of W of R^3 a subspace?

Explanation / Answer

Yes, Proof ---------- a) W is closed under vector addition ----------------------------------------------- let u = [x1 x2 x3] in W and v = [y1 y2 y3] in W hence x1 + x2 = 0 ---(1) y1 + y2 = 0 ---(2) (1) + (2) => (x1 + x2) + (y1+y2) = 0 -- (3) consider u+v, u+v = [x1+y1 x2+y2 x3+y3] hence u+v belongs to W[from 3] therefore, for every u,v in W u+w is always in W hence W is closed under vector addition b) W closed under scalar multiplication -------------------------------------------------- let u = [x1 x2 x3] in W, and k be any scalar to check , ku in W ku = [kx1 kx2 kx3] u in W => x1 + x2 = 0 k(x1 + x2) = 0 kx1 + kx2 = 0 ---(4) hence ku = [kx1 kx2 kx3] in W, (from 4) for every u in W, ku in W where k any scalar Hence W is closed under scalar multiplication from (a) and (b) W is a subspace

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