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 = b_{1}+b_{2}+2b_{3} so that [u]_{B} = (1,1,2)^{T}.
[u]_{C} = P_{C,B}[u]_{B} = P_{C,B}( 1,1,2)^{T} = (1/2,1,1/2).^{T}
(b). P_{B,c} = (P_{C,B})^{-1} =
0 |
0 |
2 |
-1 |
1 |
1 |
1 |
1 |
1 |
( c). Since P_{C,B} =
0 |
-1/2 |
1/2 |
-1/2 |
1/2 |
1/2 |
1/2 |
0 |
0 |
Hence, b_{1} = 0.c_{1}-(1/2)c_{2} +(1/2)c_{3} = -(1/2)(1,2,0) +(1/2)(1,0,0) = (0,-1,0), b_{2} = -(1/2).c_{1}+(1/2)c_{2} +0c_{3} = -(1/2)(1,2,3)+(1/2)(1,2,0) = (0,0,-3/2) and , b_{3} = (1/2)c_{1}+(1/2)c_{2} +0c_{3} = (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 P_{S, B} =
0 |
0 |
1 |
-1 |
0 |
2 |
0 |
-3/2 |
3/2 |
b_{1} =(0,-1,0), b_{2} = (0,0,-3/2) and , b_{3} =(1,2,3/2).
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