PROBLEM 5.(a) Show that S C W is a vector subspace of the vector space W, if and
ID: 3136693 • Letter: P
Question
PROBLEM 5.(a) Show that S C W is a vector subspace of the vector space W, if and only if, for all , e R and every s,te S, As + /tes. (b) Given the vector space Rt of triples of real mumbers with the usual addition and scalar multiplication, which of the following subsets make a vector subspaces PARITHA PRATIM GHOSH (c) Given the vector space Riz] of all polynomial functions with their usual addition and sealar raultiplieation which of the followiag subiets make a vector subspace of RI) In eachy of the cases also evaluate spau[SExplanation / Answer
5. (a).Let S W and let , R and s, t S, s+t S. Then assuming both , =1, s+t S s, t S. Thus, S is closed under vector addition. Also, assuming =1, = 0, s S s and R. Hence S is closed under scalar multiplication. Also 0.s = 0 apparently S. Hence S is a vector space, and, therefore, a subspace of W.
Let S W and let S be a subspace of W. Also, let , be 2 arbitrary scalars in R and let s,t be 2 arbitrary vectors in S. Since S is a vector space, it is closed under scalar multiplication. Therefore, s and t S. Further, since S , being a vector space, is closed under vector addition, hence s+t S.
(b).(i). The given set S is a vector subspace as (0,y,z)+(0,a,b) = (0,y+a,z+b) S. Thus, S is closed under vector addition. Also, k(0,y,z) = (0,ky,kz) S. Thus, S is closed under scalar multiplication. Also the zero vector (0,0,0) apparently S.
(ii). The given set S is not a vector subspace as it is not closed under vector addition. (0,y,z)+(a,0,c) = (a,y,c+z) S.
(iii). The given set S is a vector subspace as (x,-x, z)+(a,-a,c) =(x+a,-x-a,z+c). Thus, S is closed under vector addition. Also, k(x,-x,z) = (kx,-kx,z). Hence S is closed under scalar multiplication. Also the zero vector (0,0,0) apparently S.
(iv). The given set S is a vector subspace . (x, 1-x,z) +(1-y,y,z) = (x-y+1, 1-x+y, 2z) and (x-y+1)+(1-x+y) = 2 1. Hence S is not closed under vector addition.
Please post problem 6 again separately.
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