Question

Let V, W, and U be three finite dimensional vector spaces over R and T:V + Wand S : W → U be two linear transformations.

Answer 1

T: VW © and si W-tu To show in rank ISOTI > rank/Tlt rankis) adim w Proof By Rank-Nullity Theme rankl Tl + dim i Kerl T)) - dim v hankcs) & dim (Keris)) - dimw aliis rank(Sot) + dim (Ker(sot)) = dím v liii) i and then subtracting rilis Ond (11) Adding The gives rankrti & rankiss-rank/Sot) + dim ( KealT)) + dim / kerisi) -dim (KerlSot)) dimW Let {v, ug, ... ve} be basis for for Kerltl Then (S0T) (Vi) = 0 for each ľ so Ker(T) ckerl Sots Thus we Can extend this to this to a basis {v, --- Ve Y, , ,~- Yk} for Keriso T). Then for each J=1,2 J=1,2, --.-K We have O a (507) (44) s( Tl9j1)

Hence Tly ;) E Karlss for each j. further if o་ ད ag, .. ak EIR are such that a, Try, it -tak Tly, e) = o Then Tlay,+ + 9K UK ) 30 so 9.4, + takyk E Ker IT) so there are 6., bg,nor.be ER such that a, 9, + takyk = b.v. + theve => any, to -takyk-bivi --beve But a basis basis for Kerl Sot These vectors form Particular So in a, - ak co Thus a linearly {T(",).... Tlak)} is Independent Subset of of Kellsi and So dim (Keris), k Hence dim (KerlT)) + dim (Ker (si) ? dim (Kerlsot))

and So Yields the equation above ronkiti + rankis) - rank (sot) < dimlwy rankft) & rankis) dim/w) < rank (Soto Hence rank/Soto > rank/T) tronk(s) - dim w