I got part A I just need help with part B. i,ii,iii Please and thank you (a) Let
ID: 3169060 • Letter: I
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I got part A I just need help with part B. i,ii,iii Please and thank you
(a) Let V be the set of real-valued functions that are defined at each r in the interval (-0, 0). If f = f(z) and = g(x) are two functions in V and k is any scal ar, we define the operations of addition and scal ar multiplication by (f+g)(x) = f(x) + g(1), (kf)(x) = kf(z). Verify the Vector Space Axioms for the given set of vectors. (b) Given pi =1+ 2x +?, p2 = 2 + 9x, and p3 = 3 + 3x + 47°, i. show that the set S = {P1, P2, P3} is a basis for P2. ii. express p= 2 + 171 - 37° as a linear combination of the vectors in S. iii. find the coordinate vector of p relative to S.Explanation / Answer
(b). Let A =
1
2
3
2
9
3
1
0
4
It may be observed that the entries in the columns of A are the scalar multiples of 1 and the coefficients of x and x2 in p1, p2 and p3.
(i). To show that S is a basis for P2, we will reduce A to its RREF as under:
Add -2 times the 1st row to the 2nd row
Add -1 times the 1st row to the 3rd row
Multiply the 2nd row by 1/5
Add 2 times the 2nd row to the 3rd row
Multiply the 3rd row by -5
Add 3/5 times the 3rd row to the 2nd row
Add -3 times the 3rd row to the 1st row
Add -2 times the 2nd row to the 1st row
Then the RREF of A is I3. It implies that S is a basis for P2.
(ii). Let B =
1
2
3
2
2
9
3
17
1
0
4
-3
It may be observed that the entries in the first 3 columns of B are same as in A and that entries in the rth column of B are the scalar multiples of 1 and the coefficients of x and x2 in p. Then, after the same row-operations as above, the RREF of B is
1
0
0
1
0
1
0
2
0
0
1
-1
It implies that p = p1+2p2-p3.
(iii). From the RREF of B, it is apparent the coordinates of p relative to S are (1,2,-1).
1
2
3
2
9
3
1
0
4
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