Math 375 - 10/16

Question 1 • Given ○ Let V be a set ○ Let S,T:V→V be invertible functions • Prove ○ ST is also invertible and (ST)^(−1)=T(−1) S^(−1) • Proof ○ (ST)(T^(−1) S^(−1) )=S(TT^(−1) ) S^(−1)=SIS(−1)=SS^(−1)=I ○ (T^(−1) S^(−1) )(ST)=T^(−1) (S^(−1) S)T=T^(−1) IT=T^(−1) T=I Question 2 • Given ○ Let V and W be finite-dimensional vector spaces. • Proof ○ There exists a surjective linear map f:V→W if and only if dim⁡W≤dim⁡V