2cdVz0-02

2cdVz0-02 - ASSIGNMENT 2 SOLUTIONS MAT 473 SPRING 2012...

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ASSIGNMENT 2 · SOLUTIONS MAT 473 · SPRING 2012 Exercise 3.2. (i) Prove that if S GL ( n ), and T L ( R n )issuchthat ° T S ° < 1 ° S 1 ° , then T GL ( n ). (ii) Explain how it follows from this that GL ( n )isanopensubseto f L ( R n ). Proof. For (i), apply Proposition 3.2 to T S 1 : ° T S 1 Id ° = ° ( T S ) S 1 °≤° T S °° S 1 ° < 1 , so T S 1 is in GL ( n ), and it follows that T = T S 1 S is too. For (ii), part (i) shows that for each S GL ( n ), the open ball { T L ( R n ) T S ° < ° S 1 ° 1 } is contained in GL ( n ). Thus GL ( n )isopen . ° Exercise 3.4. Use Lemma 3.6 to prove that if f,g : U R n R m are di±erentiable at x U ,then ( ± f,g ² ) ° ( x )( h )= ± f ° ( x )( h ) ,g ( x ) ² + ± f ( x ) ,g ° ( x )( h ) ² for all h R n ,wherebyde²n it ion ± f,g ² ( y )= ± f ( y ) ,g ( y ) ² for all y U . Proof. Choose s , W s for f and t , W t for g as in Lemma 3.6: so W s and W t are open neighborhoods of 0 in R n ,and s : W s R m and t : W t R m are functions such that f ( x + h )= f ( x
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2cdVz0-02 - ASSIGNMENT 2 SOLUTIONS MAT 473 SPRING 2012...

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