Question 11 (10 marks) Consider two bases for P2 given by (a) Calculate the tran
ID: 3185232 • Letter: Q
Question
Question 11 (10 marks) Consider two bases for P2 given by (a) Calculate the transition matrix Pcs (which converts C-coordinates to B-coordinates) (b) Calculate the transition matrix Ps.c (which converts B-coordinates to C-coordinates). (c) Determine the polynomial p ? P2 that has the coordinate vector l| -| 3 (d) Find the coordinate vector lic for q = 1 +2-3 -4 (e) Differentiation defines a linear transformation T : P2 Pa where T(p(x))-r(x). Find the matrix of T with respect to (i) the basis B (ii) the basis C.Explanation / Answer
(a). The transition matrix PC,B =
-1
0
2
0
-4
0
5
0
0
It may be observed that the entries in the columns of PC,B are the scalar multiples of 1 and the coefficients of x, x2 in the vectors in C.
(b) Let M =
-1
0
2
1
0
0
0
-4
0
0
1
0
5
0
0
0
0
1
The RREF of M is
1
0
0
0
0
1/5
0
1
0
0
-1/4
0
0
0
1
½
0
1/10
Hence PB,C =
0
0
1/5
0
-1/4
0
½
0
1/10
( c) The polynomial p is -4(5x2-1)+3(-4x)+11(2) = -20x2-12x +26.
(d) Let N =
-1
0
2
1
0
-4
0
2
5
0
0
-3
The RREF of N is
1
0
0
-3/5
0
1
0
-1/2
0
0
1
1/5
Hence, [q]C = (-3/5,-1/2,1/5)T.
(e).(i). We have T(1) = 0, T(x) = 1 and T(x2) = 2x. Then, the matrix of T with respect to the basis B is
0
1
0
0
0
2
0
0
0
(ii). We have T(-5x2 -1) = -10x , T(-4x) = -4 and T(2) = 0. Then, the matrix of T with respect to the basis C is
0
-4
0
-10
0
0
0
0
0
-1
0
2
0
-4
0
5
0
0
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