## Answers

Sol et vand w are complementary subspace of a then VoW= R" RP: V+W and le Vow= 903 6t P= Puw be the projection of R on to u with respect to w then have for each XER" , we %= otw where ů EU and Û EW plâl= II Take s be any subspace of R Plotw) 67 X= po les3 then 6t X for some weV,wEW and ñES. then iso ما = (دا+1)P ( x ) = P Now and IL TT ve TJO T ă= ürú ES ńEW U= ă-ü ei stw (s+W) nu ie. ý € (sth)nu P(x) E (5+W)nu X = 9P (50) lates3 & (stw) ov Take Ž E (Stw)nu then Ž E stw and Ž EU ㅋ 2 - Ž = 5+ W where šES, WEW Enversely Ž-W с X ㅠ ㅠ = P() = Plz-W) = Ž P(5)= ¿ E X (s+w) ou X = { P (2) € 53 = (stW)nu T

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