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101st_tut4

# 101st_tut4 - R 2 R 3 L x = x 1 x 2 1 T(b L R 2 R 3 L x =(0...

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MATH1111/2010-11/ Tutorial IV 1 MATH1111 Tutorial IV 1. Given three ordered bases for R 3 : E = [ e 1 ; e 2 ; e 3 ], F = [ f 1 ; f 2 ; f 3 ] and H = [ h 1 ; h 2 ; h 3 ], where e 1 ; e 2 ; e 3 form the standard basis, and f 1 = (1 1 1) T ; f 2 = (1 2 2) T ; f 3 = (2 3 4) T ; h 1 = (3 2 5) T ; h 2 = (1 1 2) T ; h 3 = (2 3 2) T : (a) If [ x ] F = (1 ± 1 2) T , ±nd [ x ] E and [ x ] H . (b) Find, by evaluating [ f 1 ] H , [ f 2 ] H , [ f 3 ] H , the transition matrix from F to H . (c) Find the transition matrix from F to H by ±rst evaluating the transition matrices from F to E and from H to E . 2. What can you tell about the matrix A if A is invertible? Justify what you say. 3. Let A be an m ² n matrix where m ³ n and rank( A ) = n (i.e. A is of full rank). Denote the columns of A by a 1 ;a 2 ; ´´´ ;a n . Must A T a 1 ;A T a 2 ; ´´´ ;A T a n form a basis for R n ? Justify your answer. 4. Let F be a nonsingular 4 ² 4 matrix whose last two rows are (2 0 0 9) and (1 2 1 8), i.e. F = 0 B B @ µ µ µ µ µ µ µ µ 2 0 0 9 1 2 1 8 1 C C A : Suppose P is an 4 ² 3 matrix and FP = U where U = 0 B B @ 1 3 5 0 1 ± 1 0 0 0 0 0 0 1 C C A : Find a basis for N ( P T ).

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MATH1111/2010-11/ Tutorial IV 2 5. Determine whether the following are linear transformations. (a) L :
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Unformatted text preview: R 2 ! R 3 , L ( x ) = ( x 1 x 2 1) T . (b) L : R 2 ! R 3 , L ( x ) = (0 0 x 1 x 2 ) T . (c) L : C [0 : 1] ! C [0 ; 1], L ( f ) = F where F ( x ) = R x f ( t ) dt and C [0 ; 1] is the vector space of all continuous functions on the interval [0 ; 1]. 6. Is it possible to construct a linear transformation ful±lling the following requirements? Describe the linear transformation if yes, or give an explanation if no. (a) T : R 2 ! R 2 , T ((1 0) T ) = (2 0) T , T ((0 1) T ) = (0 2) T , T ((1 1) T ) = (2 ± 2) T . (b) T : R 2 ! R 2 , T ((1 ± 1) T ) = (2 0) T , T ((2 ± 1) T ) = (0 2) T , T (( ± 3 2) T ) = ( ± 2 ± 2) T . (c) T : R 3 ! R 2 , T ((1 ± 1 1) T ) = (2 0) T , T ((1 1 1) T ) = (0 2) T . Find their standard matrix representations for the cases of linear transformations....
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101st_tut4 - R 2 R 3 L x = x 1 x 2 1 T(b L R 2 R 3 L x =(0...

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