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Assignment sheet. III - x x x x x T-= Q.7 If T V 3 → V 3...

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BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI II Semester 2005-2006, Math C 192 Mathematics II Assignment sheet for Quiz III 13 Feb 2006 Q.1 Find a linear transformation T : V 3 → V 3 such that ker (T) = { (x,y,z) : 3x-y+z=0 & x-y+z=0}, justify your answer ? Q.2 Find a linear transformation T : V 3 → V 3 such that ker (T) = { (x,y,z) : x-y+3z=0 & 3x+y+z=0}, justify your answer ? Q.3 Find a linear transformation T : V 3 → V 3 such that R (T) = { (x,y,z) : 3x-y+z=0 }, justify your answer ? Q.4 Find a linear transformation T : V 3 → V 3 such that R (T) = { (x,y,z) : x-y+z=0}, justify your answer ? Q.5 Find a basis of ker(T) and r(T) for the linear transformation T: V 4 → V 3 defined by ( 29 ( 29 4 2 2 1 3 1 4 3 2 1 , , , , , x x x x x x x x x x T - - - = . Q.6 Find a basis of ker(T) and r(T) for the linear transformation T: V 4 → V 3 defined by ( 29 ( 29 1 3 3 1 1 4 3 2 1 , , , , , x x x
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Unformatted text preview: x x x x x T--= . Q.7 If T : V 3 → V 3 is the linear transformation defined as T(1,0,0)= (-1,0,0) , T(0,1,0)= (0,0,-1) ,T(0,0,1)= (0,1,-1) . Prove T-1 exist. Find T-1 : V 3 → V 3 Q.8 If T : V 3 → V 3 is the linear transformation defined as T(1,0,0)= (0,-1,0) , T(0,1,0)= (0,0,-1) ,T(0,0,1)= (-1,0,1) . Prove T-1 exist. Find T-1 : V 3 → V 3 Q.9 Page 153 Q1 (a) Q.10 Page 157 Q2(b) 13. Find a basis of N(T) and a basis of R(T) of the linear transformation T: V 3 → V 3 , defined by ( 29 ( 29 , 3 2 , , , 3 2 1 3 2 1 x x x x x x T--= . Hence find n(T) and r(T). Page 193 Q.1(a), (c) (f) Q 5 (b) Page 200 Q.1 (a) , (c), (f), (g) Page 202 Q.2 Page 68 Q.6 ( c) , (g) Page 73 Q.4 (e) , Q.5 (d) Page 76 Q. 1 ( c ) , Q.2 (b)...
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