Question

Q9. Let W be a subspace of R". (a) Prove that w+ is a subspace of R". (b) Prove that if a vector v belongs to both W and W+, then v must be the zero vector.

Answer 1

Since we are in R^n we can talk about innerproduct. To prove a set is a subspace we need only show that the set is closed under vector addition and scalar multiplication induced from R^n.

w is Given subspace of R". a) To prove wt is subspace of R". Since <o,o) = wt and w his non-empty Let 0 Hoew, x. ye wat Then <%,19) = 0 & vin w Ty.ws tv in w. Now <xty, v) <x,0) + <y, v> Noin w. Thus E xxy wt. Now for CER, wt, vew <ex, us c<x,0) XE - & cer, xewt and wew. Thue <cx, v) = 0 So cx e w subsface of R. So wt is a

for wew 19 см and we b) Let ve wand vewt, we need to prove v=o. Ewt <0,0) = (consider first & from w and secorid v from wt) V=0 (by the property of inner product <«,«>=0 <=> x = x =o). So v is the vector. Zero