Question

31. Find an isomorphism L: R → R".

Answer 1

we prove need to L: Rn Rh is isomorphism Let al L([a, az az az. an) Let us verify one to one consider two vertors [a, az az an] & [b, bz bz. be such that Lai az...am EL [bs by 13.,.ba 1 92 93 b b2 b3 using a3 bn an which means Therefore, the one-to-one az = 52, 93=bz- is a = be, mapping Lily MR vexity for Linear to ansformation Consider can [ by bynabn a2 93--- an ¢ let us are two vectors in Rn

Then 4[a, az 93.-9n] + [6, bx bp.-.be] ([a, tb, aztbe aztbz antha L Tathi = azt bz from aztb3 ant bo which can be written as b) 7 a L[a, a an] + [b, bz ... bn = from so linear it is Let [a, a constant an] ERK, and k is some L(k[a, a ...an]) = ([ko, kes kan kai Kar K 92 an Kan an 92 so L(k[a, q ..an KL[9,

so therefore Lirneh is linear Transformation verify onto Rh By definition we have Lirn- Llia, az ....9 al an り R there is This means for every Lan [ a 92 Lorn an Ern under LiRory So by definition of onto, is onto This Lilor? is me-to-one onto of linear transformation Therefore, I is an is omorphism from Ro Rh

Plz do Upvote