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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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 Fall '07
 Schilling
 Linear Algebra, Algebra, Matrices

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