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A university examination from the applied linear algebra course (mathematics 307) held at the university of british columbia in december 2010. The examination covers various topics such as matrix norms, eigenvalues, polynomial interpolation, least squares fits, fourier series, and recursive sequences. Students are required to solve problems related to finding eigenvalues, sketching graphs, writing down basis vectors, and determining coefficients.
Typology: Exams
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Marks
[15] 1. Let
A =
0 a 0 0 0 2
where a is a real number. (a) [4] For what values of a (if any) does the matrix norm have the value ‖A‖ = 2?
(b) [2] For what values of a (if any) is cond(A) not defined? Give a reason.
(c) [2] For what values of a (if any) is cond(A) = 1/2? Give a reason.
(d) [4] Sketch a graph of cond(A) as a function of a for −∞ < a < ∞.
(e) [3] For what values of a (if any) is cond(A) = 4?
[12] 3. Suppose we are given 4 points (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) and (x 4 , y 4 ) in the plane and we want to find a function f (x), defined for x 1 ≤ x ≤ x 4 , whose graph interpolates these points. Assume that
f (x) =
p 1 (x) for x 1 ≤ x ≤ x 2 p 2 (x) for x 2 ≤ x ≤ x 3 p 3 (x) for x 3 ≤ x ≤ x 4
where each pi(x) is a polynomial.
(a) [3] What equations, written in terms of pi(x) and possibly their derivatives, express the condition that f (x) goes through the given points? Do these equations imply that f (x) is continuous?
(b) [3] What equations, written in terms of pi(x) and possibly their derivatives, express the condition that f ′(x) is continuous?
(c) [3] What equations, written in terms of pi(x) and possibly their derivatives, express the condition that f ′′(x) is continuous?
(d) [3] When each pi(x) is a cubic polynomial of the form ai(x−xi)^3 +bi(x−xi)^2 +ci(x−xi)+di the equations written in parts (a), (b) and (c) above are equivalent to a system of linear equations in the unknowns ai, bi, ci and di, i = 1, 2 , 3. How many more equations are needed if there are to be the same number of equations as unknowns? What equations are usually added and why?
[10] 4. In this question we are once again given 4 points (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) and (x 4 , y 4 ) in the plane. This time we want to find a quadratic function q(x) = ax^2 + bx + c that comes closest to going through the points by doing a least squares fit.
(a) [5] The least squares equation you need to solve to find the coeficients a, b and c has the form AT^ Aa = AT^ b. Write down expressions for A, a, and b.
(b) [5] Suppose the points (xi, yi) have been defined in MATLAB/Octave as X1,.. ., X4, Y1,... Y4. Write down the MATLAB/Octave code that plots these points, then com- putes q(x), and finally plots q(x).
(b) [5] If the matrix A from part (a) is defined in MATLAB/Octave, we can do the following calculations:
eig(A) > abs(eig(A)) ans = ans =
1.83929 + 0.00000i 1. -0.41964 + 0.60629i 0. -0.41964 - 0.60629i 0.
Describe how you could make further use of the eig command and other MATLAB/Octave com- mands to determine all (possibly complex) initial values a, b and c for which xn → 0 as n → ∞.
(c) [3] Explain how you could ensure that the a, b and c you find in part (b) are real numbers.
(a) [3] Determine the coefficients cn in the expansion f (x) =
n=−∞ cne
2 πinx, where f (x) = x and 0 ≤ x ≤ 1.
(b) [3] Calculate the inner product 〈f (x), f (x)〉 for f (x) = x on the interval 0 ≤ x ≤ 1, using the definition of the inner product for functions.
(e) [3] Explain how you could use the fft command in MATLAB/Octave to compute ap- proximations to the coefficients cn in part (a). Write down the commands you would use, and say for what values of n you would expect your approximations to be most accurate.
(f) [3] Suppose you expanded the same function f (x) = x as in part (a) except on the interval 0 ≤ x ≤ 2. What would be the form (i.e., do not compute the coefficients) of the Fourier series valid for this interval. What points on the plane would you plot to produce a frequency-amplitude plot from this new Fourier series? (Give the answer in terms of the coefficients in the new expansion.)
[15] 7. Suppose A is a symmetric 4 × 4 matrix with eigenvalues 0, 1 , 4 , 5. Define a sequence of vectors xn ∈ R^4 by choosing x 0 at random, and then setting
yn = (A − 3 I)−^1 xn− 1 xn = yn/‖yn‖
for n = 1, 2 ,.. .. You then observe that xn converges to x∞ = [1/ 2 , 1 / 2 , 1 / 2 , 1 /2]T^ as n → ∞.
(a) [0] What is Ax∞?
(b) [0] What is the value of the inner (dot) product 〈x∞, Ax∞〉?
(c) [0] What vector does yn converge to?
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