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....
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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.
 Fall '07
 Schilling
 Linear Algebra, Algebra, Matrices

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