Question

Problem 1: Let X be a linear space. Let Y CX be a linear subspace. (a) Prove that the map : X+X/Y given by 7(x) = (2) is a linear map. (b) Prove without using the dimension formula or rank-nullity that N = Y. (c) Prove without using the dimension formula or rank-nullity that RX = X/Y.

Answer 1

let 1. Let, x be a linear space and YC X be a linear subspace. la TixXly given by (m) = [»] Here [x] = x+Y. het, n, gtx and a be a scalar. " T(m+) = (x+] = (2+a) +Y = (x+Y) + (+Y). = [x] + [3] = T(m) + (3) TT (**) = [dn] = 0m+y = x (m+Y) =d[] =dTM (m+2) = () + () for all megex (2x) = 2T (n) for all ntx and for all so, π is a linear map (b) NT - { x + x | = -) = []} . Let, at na ...Bo, T(M)= [o] on, at y = 0+Y or, nEY reald. So, Naçi

het, mE Y T(m) = m+Y R = 0+Y [an nty = [0] soint na 1. Therefore YC NA -. No=y (C) R = {x | y = 15 (m2 for all sex} . Clearly RH C X/Y . }, [] X/Y . Then there exists net such that TT (n) = [n]. so, π is a su map. .: R= x/y.