Question

(2 points each for (a),(b),(d)) In this problem, we will prove the following di- mension formula. Theorem. If H and H' are subspaces of a finite-dimensional vector space V, then dim(H+H') = dim(H)+dim(H') - dim(H nH'). (a) Suppose {u1;...; up} is a basis for H OH'. Show that we can find vectors V1, ..., Vi € H and ví, ...,V' € H' such that {u1; ...; Up; V1;...; Vi} is a basis for H and {ui;...; Up; vá;...;V;} is a basis for H'.! (b) Let B = {u1; ...; Up; V1;...; Vi; v1;...;v;} be the union of the two lists in part (a). Show that span(B) = H + H'.2 (c) (2 bonus points) Show that B is linearly independent.3 (d) Use the results of parts (a), (b), and (c) to prove the theorem.

1Hint: Use the theorem from class that any linearly independent list of vectors is contained in a basis

2Hint: Remember that we prove the equality of sets X = Y by showing X ⊂ Y and Y ⊂ X.

Answer 1

span {B} (3) Supplementary explanations are given below :- (a) Note that {4,4,-, up} is a linearly independent sek contained in Hi E can be extendended to bases (say) By = {1, 2, - , Up, Y, VZ, as Vi} of Hi By similar argument we get E can be entended to a a basis (say, B = {4, he .. up , vi, ves of the (b). Note that B={1, 42, - , Up, Video - Va, vi , L, -, vi} = B,UBE . Hie span (Be) ç span (B) and H₂ ç span (B₂) s span (B) Hitta s span (B) leo we here span (B;) Chi span (B) = 4₂ "H=span (B) & span {B, UBe} = span {3} + span {B} = H, THz -622 (1) 8 (2) ² Hit H₂ (e), suppose of Up + 1/4 + .. up up + B, Vit Beet + ßirit =ū for some scalars, h, t... Bella Batz to. Bivi + B, vt lukt. Bliv Tp up ut, a ve vlak, on, ut do not belong to the sub space HnH bul -44 - 42 " .. EHINH and hence the only possibility is both rectors are zero vectors. .:B d=0,42=0,--, 420 Bir + PV +.- -B'v' - -. - BIG . Both vectors are zero vectors since then, Vre 4-694 and isliklare vile H- HAH and (H - HOH)n (H- HAH') = 0 , and hence the only possibility is both vectors are puthu + 8; v: 20 = Alult ... + Blv = 120, 150,--, pizo and .'=0,12=0, since B , B are linearly independent sete. linearly independent (d). From (3) and (4) we get B is a .! chim (Het Hu) = þ + i + j = ..(pri) + 8 + j) - f • olim (H) + dim (H') - olim CHOH'). = -44, -de 4₂ so eince zero vectos. 2 Thus B is set (42 basis for H, + H₂