jUIfHt-03

JUIfHt-03 - 6 SPRING 2012 3 Invertible Operators Derivatives For each T L(Rn let det T be the determinant of the n n matrix representing T with

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6 SPRING 2012 3. Invertible Operators, Derivatives For each T L ( R n ), let det T be the determinant of the n × n matrix representing T with respect to the standard basis; this de±nes a map det: L ( R n ) R . In fact, det is continuous. To see this, note that the map de±ned by T °→ ( a 1 , 1 ,...,a n, 1 ,a 1 , 2 ,...,a n, 2 ,...,a 1 ,n ,...,a n,n ) whenever [ T ]=( a ij ), is a normed vector space isomorphism of ( L ( R n ) , ±·± 2 ) onto the Euclidean space R n 2 . But viewed as a function on R n 2 , det is just a n th -degree polynomial in n 2 variables, and is therefore continuous. Defnition 3.1. For each n N ,let GL ( n ) be the set of invertible elements of L ( R n ). Arguments similar to the one made above for det (and Cramer’s formula for inverses) show that the maps ( S,T ) °→ S T from GL ( n ) × GL ( n )to GL ( n )and T °→ T 1 from GL ( n )to itself are continuous. Thus GL ( n )isa topological group , called the general linear group . Exercise 3.1. Prove or disprove: ± T 1 ± = 1 ± T ± for each T GL ( n ) . Moreover, since GL ( n ) is the inverse image of the open set R \{ 0 } under the continuous map det, it is an open subset of L ( R n ). In other words, for each invertible S L ( R n ), there exists °> 0 such that all T L ( R n ) which satisfy ± S T ± are also invertible. It turns out that with S = Id, we can take ° = 1 (think of the case n =1) . Proposition 3.2. For T L ( R n ) , if ± Id T ± < 1 , then T
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JUIfHt-03 - 6 SPRING 2012 3 Invertible Operators Derivatives For each T L(Rn let det T be the determinant of the n n matrix representing T with

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