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a=0 b=0 c=6 2 3 2 Consider the function : M3 - My defined by L(4)...
# 2 and # 3
2 -6 4 -4 0 -4 6 1. Define A = 8 01 . Determine, by hand, the LU factorization, of A. You may of course check your answer using appropriate technology tools. Then use your result to solve the system of equations Ax b, where b--4 2 0 5 2 2. Suppose A-6 -3 133Even though A is not square, it has an LU factorization A LU, 4 9 16 17 where L and...
13 please
8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B...
Applied Mathematics Laplace Transforms
1. Consider a smooth function f(t) defined on 0 t<o, with Laplace transform F(s) (a) Prove the First Shift Theorem, which states that Lfeatf(t)) = F(s-a), where a is a constant. Use the First Shift Theorem to find the inverse trans- form of s2 -6s 12 6 marks (b) Prove the Second Shift Theorem, which states that L{f(t-a)H(t-a))-e-as F(s), where H is the Heaviside step function and a is a positive constant. Use the First and...
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a + (b + c)2 + dx?. (a) Show that L is a linear transformation, that is, show that L(sA+B) = 8L(A) + tL(B) for any A, B E M2x2(R) and any st ER. (b) Consider p(x) = ao+a1x + a2x2 € P2(R). Show that L is onto by showing that L(A) = P(x) for some matrix A € M2x2(R). Note that you must give...
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
Consider the function φ : R- (a) What is ф(6)? (b) What is(-3)? (c) What is φ(R)? (d) What is 0-1 (16)? (e) What is φ-1 (1/4)? (f) What is ф_1(2, 32)? R defined by ф(x)
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks
3. Consider the function defined by...
3 2 1 1 2 3 3) Let C- 2 6-1and D 0 5 6 0 09 12 0 a) Find det(C) b) Find det (D) c) Find det (CD) d) Find det(DC)
2 1 -2 3 0 1 4 2 1. Let B -3 0 3 ( 1) 2 2 -1 0 (a) Find det(B).(Show all work.) -3 -R2- .A 4 O0-2/2 1-3 0 3 入ス-1 0 I-2 3 det ao -1 O 3 1-3 RyR-( 2 2-10 420 4 (b) Find det(BT). (c) Find det(B-1). (d) Find det(-B) . (e) Is 0 an eigenvalue of B? (f) Are thè columns of B linearly independent?