101st_tut5sol

101st_tut5sol - MATH1111/2010-11/Tutorial V Solution 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH1111/2010-11/Tutorial V Solution 1 Tutorial V Suggested Solution 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 . Ans . (a) To show the linearity, we check the following: S ± T ( x + y ) = S ± T ( x + y ) ² = S ± T ( x )+ T ( y ) ² = S ± T ( x ) ² + S ± T ( y ) ² = S ± T ( x )+ S ± T ( y ). S ± T ( ± x ) = S ± T ( ± x ) ² = S ± ±T ( x ) ² = ±S ± T ( x ) ² = ±S ± T ( x ). (b) Let x 2 V . By de±nition, S ± T ( x ) = S ( T ( x )). Thus, [ S ± T ( x )] E = [ S ( T ( x ))] E = A [ T ( x )] E = AB [ x ] E : Using the uniqueness of matrix representation, we conclude that AB is the matrix repre- sentation.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
MATH1111/2010-11/Tutorial V Solution 2 2. Let T be a linear operator on R 3 de±ned 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 . Show that this matrix representation is nonsingu- lar. Ans . (a) The standard matrix representation is 0 @ 3 0 1 ± 2 1 0 ± 1 2 4 1 A , which is nonsingular by checking its determinant. (b) From 0 @ 3 0 1 ± 2 1 0 ± 1 2 4 1 A ± [ u 1 ] St [ u 2 ] St [ u 3 ] St ² = 0 @ 3 0 1 ± 2 1 0 ± 1 2 4 1 A 0 @ 1 ± 1 2 0 2 1 1 1 1 1 A = 0 @ 4 ± 2 7 ± 2 4 ± 3 3 9 4 1 A ( St denotes the standard ordered basis for R 3 ), we see that as x = [ x ] St in R 3 , T ( u 1 ) = [ T ( u 1 )] St = (4 ± 2 3) T ; T ( u 2 ) = ( ± 2 4 9) T ; T ( u 3 ) = (7 ± 3 4) T : Then ± [ T ( u 1 )] E [ T ( u 2 )] E [ T ( u 3 )] E ² = 0 @ 1 ± 1 2 0 2 1 1 1 1 1 A ± 1 0 @ 4 ± 2 7 ± 2 4 ± 3 3 9 4 1 A is the matrix representation w.r.t E . 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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

101st_tut5sol - MATH1111/2010-11/Tutorial V Solution 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online