082nd_tut9sol

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

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Unformatted text preview: MATH1111/2008-09/Tutorial IX Solution 1 Tutorial IX Suggested Solution 1. Recall: If f : A → B and g : B → C are two functions and B ⊂ B , 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 ....
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082nd_tut9sol - MATH1111/2008-09/Tutorial IX Solution 1...

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