# Homework_4.pdf - Homework 4 Problem 1 Suppose R R2 u2192...

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Homework 4 Problem 1. Suppose R : R 2 R 2 is counter-clockwise rotation by π/ 2. Does { R, R 2 , R 3 , R 4 } form a basis for L ( R 2 , R 2 )? Problem 2. Find all possible signatures that can be realized by a nilpotent map on R 4 . If it’s possible, find an example. If not, prove that it’s impossible. Problem 3. A trace is a non-zero linear functional T : M n ( R ) R such that T ( AB ) = T ( BA ) for all A, B M n ( R ). Suppose τ : M 2 ( R ) M 3 ( R ) R satisfies the property τ ( AB ) = τ ( BA ) for all A, B M 2 ( R ) M 3 ( R ) and that τ ( I 2 0 3 ) and τ (0 2 I 3 ) are nonzero. Show that τ is the direct sum of a trace on M 2 ( R ) and a trace on M 3 ( R ). Problem 4. Show that the derivative operator