Assignment Sheet. Iii

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 : V3→V3 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 : V3→V3 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 : V3→V3 such that R (T) = { (x,y,z) : 3x-y+z=0 }, justify your answer ? Q.4 Find a linear transformation T : V3→V3 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: V4 → V3 defined by T ( x1, x2 , x3 , x4 ) = ( x1 − x3 , x1 − x2 , x2 − x4 ) . Q.6 Find a basis of ker(T) and r(T) for the linear transformation T: V4 → V3 defined by T ( x1, x2 , x3 , x4 ) = ( x1, x1 − x3 , x3 − x1 ) . Q.7 If T : V → V is the linear transformation defined as 3 3 T(1,0,0)= (-1,0,0) , T(0,1,0)= (0,0,-1) ,T(0,0,1)= (0,1,-1) . -1 -1 Prove T exist. Find T : V → V 3 3 Q.8 If T : V → V is the linear transformation defined as 3 3 T(1,0,0)= (0,-1,0) , T(0,1,0)= (0,0,-1) ,T(0,0,1)= (-1,0,1) . -1 -1 Prove T exist. Find T : V → V 3 3 Q.9

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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: V3 → V3, defined by T ( x1 , x 2 , x3 ) = ( 0, x1 − 2 x 2 − 3 x3 ,0 ) . 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)