Unformatted text preview: T,S : V → V be linear transformations. The goal of this problem is to prove that T ◦ S is invertible if and only if both T and S are invertible. (a) Show the “easy” direction: if T,S are both invertible then T ◦ S is invertible and its inverse is equal to S1 ◦ T1 . (b) Show that range( T ◦ S ) ⊆ range( T ). Conclude that if T is not invertible (which for a linear operator we know is equivalent to not being surjective) then T ◦ S is also not surjective (and therefore not invertible). (c) Similarly, show that null( S ) ⊆ null( T ◦ S ). Conclude that if S is not invertible (which for a linear operator we know is equivalent to not being injective) then T ◦ S is also not injective and therefore not invertible....
View
Full Document
 Fall '07
 Schilling
 Linear Algebra, Algebra, Matrices, Inverse function, matrix product AB, matrix product BA

Click to edit the document details