Question

Assume that the transition matrix from basis B = {b1, b2, b3} to basis C = {c1, c2, c3} is PC,B = 1/2*[ 0 -1 1 ; -1 1 1 ; 1 0 0 ].

(a) If u = b1 + b2 + 2b3, find [u]C.

(b) Calculate PB,C.

(c) Suppose that c1 = (1, 2, 3), c2 = (1, 2, 0), c3 = (1, 0, 0) and let S be the standard basis for R 3 . (i) Find PS,B. (ii) Using part (i), determine the explicit form of the vectors b1, b2, b3.

2. (a). u = b1+b2+2b3 so that [u]B = (1,1,2)T.

[u]C = PC,B[u]B = PC,B( 1,1,2)T = (1/2,1,1/2).T

(b). PB,c = (PC,B)-1 =

 0 0 2 -1 1 1 1 1 1

( c). Since PC,B =

 0 -1/2 1/2 -1/2 1/2 1/2 1/2 0 0

Hence, b1 = 0.c1-(1/2)c2 +(1/2)c3 = -(1/2)(1,2,0) +(1/2)(1,0,0) = (0,-1,0), b2 = -(1/2).c1+(1/2)c2 +0c3 = -(1/2)(1,2,3)+(1/2)(1,2,0) = (0,0,-3/2) and , b3 = (1/2)c1+(1/2)c2 +0c3 = (1/2)(1,2,3)+(1/2)(1,2,0)= (1,2,3/2).

Let A =

 1 0 0 0 0 1 0 1 0 -1 0 2 0 0 1 0 -3/2 3/2

Hence PS, B =

 0 0 1 -1 0 2 0 -3/2 3/2

b1 =(0,-1,0), b2 = (0,0,-3/2) and , b3 =(1,2,3/2).

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