101st_tut5

101st_tut5 - MATH1111/2010-11/ Tutorial V 1 MATH1111...

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MATH1111/2010-11/ Tutorial V 1 MATH1111 Tutorial V 1. Recall: If f : A ! B and g : B ! C are two functions, then the composite f ± g of f and g is a function from A to C de±ned as f ± g ( x ) = f ( g ( x )) . (a) Let S : U ! W and T : V ! U be two linear transformations. Show that S ± T : V ! W is a linear transformation. (b) Suppose S;T : V ! V are two linear operators and let E be an ordered basis for V . If A and B are the matrix representation of S and T w.r.t. E (i.e. r.t. E and E ) respectively, show that the matrix representation of S ± T w.r.t E is AB . 2. Let T be a linear operator on R 3 de±ned by T (( x y z ) T ) = (3 x + z ² 2 x + y ² 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 = ( ² 1 2 1) T , u 3 = (2 1 1) T . Find the matrix representation of T w.r.t. E . Show that this matrix representation is nonsingu- lar. 3. Let L : V ! V be a linear operator and A be its matrix representation w.r.t. an ordered basis E . Suppose F is another ordered basis for V and det A 6 = 0. Is the matrix representation w.r.t.
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101st_tut5 - MATH1111/2010-11/ Tutorial V 1 MATH1111...

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