HW4 - AMS210.01. Homework 4 Andrei Antonenko Due at the...

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Homework 4 Andrei Antonenko Due at the beginning of the class, March 28, 2003 1. Let f : R 3 R 2 such that f ( x,y,z ) = ( x + 2 z, 2 x - y - z ) and g : R 2 R 3 such that g ( x,y ) = (2 x + y, x + 2 y, x + y ) . Find ( f g ) and ( g f ). 2. Check which of the following functions are linear. Justify your answer. (a) f ( x,y ) = ( x, y 2 ) (b) f ( x,y ) = ( | x + y | , x - y ) (c) f ( x,y,z ) = ( x + y, z, 0) (d) f ( x,y ) = (( x + 1) 2 , y + 3) (e) f ( x,y,z ) = ( x + y + z, y + z, z ) 3. Let f : M n,n R be a function which maps any n × n matrix to a sum of its diagonal elements. Determine is this function linear or not. 4. Let V be a vector space of all n × n -matrices, and M be a fixed matrix in V . Which of the following functions T i : V V are linear: (a) T 1 ( X ) = XM (b) T 2 ( X ) = X + M (c) T 3 ( X ) = XM - MX (d) If M is invertible, T 4 ( X ) = MXM - 1 5. Let f : R 3 R 2 such that f ( x,y,z ) = (2 x + y, x + y + z ). (a) Check that
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.

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HW4 - AMS210.01. Homework 4 Andrei Antonenko Due at the...

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