í Chrome File Edit View History Bookmarks People Window Help 0 0 > 56% D Tue 9:5
ID: 3116991 • Letter: #
Question
í Chrome File Edit View History Bookmarks People Window Help 0 0 > 56% D Tue 9:56 PM Q E DO. C Home | Cix % Eleventh x P MATH 23 x 0 e3.pdf x D ans.chbar x D e3ans.pdil x f (1) Pedro x C Crunchyre x A WebAssig x s Watch x Pedro - e a Secure https:/piazza-resources.s3.amazonaws.com/j6wopzg|2fx2uj/jaioabxlkolic/e3.pdf?AWSAccessKeyld=AKIAIEDNRLJ4AZKBWGHA&Expires;=151253. * 0 : 70 pts. Problem 3. Consider the space P = {az+4{z+ ao| a; ER | } } of polynomials of degree less than three. One basis of Pz is | P= [1 ?). Define polynomials by 91 () - 1, 2(z) -1-3, 93 (T) – (z - 312 - ? - 6x + 9, and define Q- [g1() q2(1) 95()]. A. Verify that Q is a basis and find the change of basis matrices Spe and Sep. B. Let f(1) = 2 + 31 + 57°. Find the coordinates (f(z)]p. C. Find the coordinates (S(0)le of f(z) with respect to the basis Q. Write f(x) as a linear combination of qi(), 92(r) and g3(*). Expand and simplify this expression to check that it's correct. D. Let g(r) = 3 - 5(- 3) 3(- 3). Find the coordinates (g(x)]p of g(1) with respect to the basis P by using the change of basis matrix. Check that this is correct.Explanation / Answer
3. Let A =
1
-3
9
0
1
-6
0
0
1
The RREF of A is I3. This implies that q1,q2,q3 are linearly independent and span P3.Hence Q= { q1,q2,q3} is a basis for P3.
Let M =
1
-3
9
1
0
0
0
1
-6
0
1
0
0
0
1
0
0
1
The RREf of M is
1
0
0
1
3
9
0
1
0
0
1
6
0
0
1
0
0
1
Hence the change basis matrix from P to Q, i.e. SPQ =
1
3
9
0
1
6
0
0
1
Also, the change basis matrix from Q to P, i.e. SQP =
1
-3
9
0
1
-6
0
0
1
B. f(x) = 2+3x+5x2.Then [f(x)]P =(2,3,5)T.
C. Let P =
1
-3
9
2
0
1
-6
3
0
0
1
5
The RREF of P is
1
0
0
-56
0
1
0
33
0
0
1
5
Hence [f(x)]Q = (-56,33,5)T. Further, -56(-1)+33(x-3)+5(x2-6x+9) = 56+33x-99+5x2-30x+45 =5x2+3x+2 = f(x).
D. g(x) = 3-5(x-3)-3(x-3)2 = 3-5x+15-3(x2-6x+9) = -3x2 +13x-9= -9+13x-3x2 so that [g(x)]P = (-9,13,-3)T.
Also, [g(x)]P = SQP.(3,-5,-3)T =(-9,13,-3)T.
1
-3
9
0
1
-6
0
0
1
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.