082nd_tut9sol - MATH1111/2008-09/Tutorial IX Solution 1...

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MATH1111/2008-09/Tutorial IX Solution 1 Tutorial IX Suggested Solution 1. Recall: If f : A B and g : B 0 C are two functions and B B 0 , then the composite f g of f and g is a function from A to C defined as f g ( x ) = f ( g ( x )). Let S and T be two linear operators on V , and E be an ordered basis for V . (a) Show that S T : V V is a linear operator. (b) If A and B are the matrix representation of S and T w.r.t. E respectively, show that the matrix representation of S T w.r.t E is AB . Ans . (a) Suffices to show the linearity. S T ( x + y ) = S ( T ( x + y ) ) = S ( T ( x )+ T ( y ) ) = S ( T ( x ) ) + S ( T ( y ) ) = S T ( x )+ S T ( y ). S T ( α x ) = S ( T ( α x ) ) = S ( αT ( x ) ) = αS ( T ( x ) ) = αS T ( x ). (b) Let x V . By definition, S T ( x ) = S ( T ( x )). Thus, [ S T ( x )] E = [ S ( T ( x ))] E = A [ T ( x )] E = AB [ x ] E . 2. Let T be a linear operator on R 3 defined by T (( x y z ) T ) = (3 x + z - 2 x + y - x + 2 y + 4 z ) T . (a) Find its standard matrix representation and show that it is nonsingular matrix. (b) Let E = [ u 1 , u 2 , u 3 ] where u 1 = (1 0 1) T , u 2 = ( - 1 2 1) T , u 3 = (2 1 1) T . Find the matrix representation of T w.r.t. E (i.e. r.t. E and E ).
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