Mathematics IB Tutorial 4
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Mathematics IB Tutorial 4
1. Let F : R4 - R3 be a linear transformation satisfying
F (1, 1, 1, 1) = (0, 1, 2), F (1, 1, 0, 0) = (0, 0, 0),
F (1, 1, 0, 1) = (0, 0, 2)
F (0, 1, 0, 0) = (_1, 0, 0)
(a) Calculate F (x, y, z, w)
(b) Calculate ker(F) and R(F).
2. For each of the following nonhomogeneous DEs, write down the appro- priate undetermined coefficients guess for the solution. (If time allows you can determine the coefficients and check them substitution).
(a) y\\ + 2y\ + y = 2t2 + 1
(b) y\\ + 3y + 2y\ = e一t + et
(c) y\\ (x) + 4y(x) = cos (2x)
(d) [Non- examinable challenge] y\\ _ 4y + 4y\ = t sin 2t
(Hint for (d): go through the thought process which helped fill in the different forms of functions given in Table 1.2 to this situ- ation.)
3. Determine whether the sequences defined as follows (for n e N) con- verge, and if they do, find the value to which they converge.
n2 _ 3n3
(a) an =
(_1)n
(b) bn = cos( n2 )
^n2 + 1
Practice Questions
Algebra
Linear Transformations
1. What are all the linear transformations R - R?
2. (a) Give the definition of a linear transformation.
(b) Decide, giving reasons, whether each of the following functions
F are linear transformations (if it is a linear transformation then show that the conditions in the definition are satisfied):
(i) F (x, y) = (y, x, x, y) (ii) F (x, y, z) = xyz (iii) F (x, y, z) = (x + y, x _ y)
A = ╱c(a) d(b)、,
where a, b, c, d e R and define the linear transformation F (v) = Av.
(a) Find conditions on a, b, c, d such that F (v) is always proportional
to v.
(b) If v and F (v) are both on the line y = mx, then they differ only
in their lengths. Determine the relationship between the length of v and F (v) for the values calculated in part (a).
4. [Non- examinable challenge] Find an example of a function F : R2 - R2 such that F (ku) = kF (u) for all k e R, u e R2 but F is not a linear transformation. (Hint: note that F maps lines to lines).
5. (a) Define the kernel and range of a linear transformation.
(b) Prove that the kernel and range are subspaces (of the appropriate
vector spaces).
(c) For each of the following linear transformations find the kernel and range.
(i) F (x, y, z) = x + y + z ,
(ii) G(x, y, z) = (y, x, y),
(iii) H(x, y, z) = (y, z, x).
6. Let F (v) = Av be the linear transformation from R5 to R3 defined by
A = ┌┐
7. A linear transformation F is said to be one-to- one if it satisfies the following condition: if F (u) = F (v) then u = v. Prove that F is one-to-one if and only if ker(F) = {d}.
8. [Non- examinable challenge] Find an example that shows that it is pos- sible that to have a function F : Rn - Rn such that F (V) is a subspace whenever V is a subspace, but where F is not a linear transformation.
9. [Non- examinable challenge] The set Z3 = {0, 1, 2} is an example of a finite field if we define addition and multiplication in such a way that we always take the remainder of the answer when divided by 3. For example 2 + 1 = 0, 2 + 2 = 1 and 2 x 2 = 1. We can then define a finite vector space such as Z3(n) which consists of vectors with n entries in Z3, and with scalar multiplication defined by using scalars in Z3 as well. All of our basic results on vector spaces still work in this setting.
(a) Find all 1 dimensional subspaces of Z3(2) .
(b) Define F : Z3(2) - Z3(2) by F (x, y) = (x+ 2y, 2x+ y). Find the kernel
and range and verify the dimension theorem. (Hint - the result should be different to that for the same function from R2 - R2 .)
Calculus
Second order constant coefficient non-homogeneous equa- tions
10. Find the general solution of the following differential equations.
(a) + 4y = 2x2 .
(b) _ y = e一2北 .
11. Solve the following initial-value problems.
(a) y\\ + 3y + 2y\ = et ; y(0) = 0 , y\ (0) = 3.
d2y dy dy(0)
dx2 dx dx
12. [Non- examinable challenge] In many vibrational problems, a forcing can be represented as the linear combination of many modes of differ- ent frequencies. In this spirit, consider a forced undamped block-spring model (or equivalently, a circuit comprising an inductor and capaci- tor connected to a voltage source), whose governing equation can be
written as
N
y\\ (t) + ω2y(t) = amsin (mπt) ,
m=1
in which ω > 0 is the system’s natural frequency. The number of modes, N , as well as the amplitudes, a1 , . . . aN , of the additional im- posed forcing/voltage, are assumed known.
(a) Assuming that ω mπ for any m = 1, . . . , N , find the general
solution to the system.
(b) How does your solution change if ω = mπ for one of the m values?
Interpret the long-term behaviour.
13. [Non- examinable challenge] Construct a second-order constant-coefficient nonhomogeneous DE for the dependent variable y(t) in which when you apply the method of undetermined coefficients to find a particular solution, you are forced to multiply your initial guess twice by t.
Sequences, Series and Convergence
Sequences
14. Write down the general term an for each of the following sequences.
(a) {1, 3, 5, 7, 9, 11, . . . } (odd positive integers)
(b) {_1, 1, _1, 1, _1, 1, _1, 1 . . . } (alternating _1 and 1s)
(c) {1, 4, 9, 16, 25, . . . } (square numbers)
(d) {1, 3, 6, 10, 15, 21, . . . } (triangular numbers)
(e) ,1, _ , , _ , , _ , . . . 、(alternating reciprocals of the count-
ing numbers)
15. Determine whether the sequences defined as follows (for n e N) con- verge, and if they do, find the value to which they converge.
n
1 + n
(b) bn = (_1)n
3 + 5n2
(c) cn =
(e) en = sin
(f) fn = sin n
16. Let a e R and d e R be constants, and consider the sequence
n _ 1
n
Show that the sequence bn satisfies the recurrence relation
d
bn+1 = bn +
n→&
limit n - o to both the original definition and the recurrence rela- tion?
17. Assuming that the limit exists, find the limit of the sequence
{an} := ,.^2, ←2^2, ~2 ←2^2, |2 ~2 ←2^2 , . . . 、.
. .
by first writing a recurrence relation for the sequence, and then taking the limit n - o.
18. We know that the standard Fibonacci sequence {an} is defined by an+2 = an+1 + an ; n e N ,
with initial conditions a0 = 1 and a1 = 1. Define bn := an+1/an, i.e., the sequence generated by taking the ratio between adjacent Fibonacci numbers. Show that if the sequence bn converges to a value b, then b = (1 + ^5)/2. [Hint: start by dividing the Fibonacci sequence definition by an+1, and writing this in terms of the bn-sequence.]
19. Suppose you take out a home loan for L = $300, 000, at an interest rate of r% per annum, which is compounded monthly. You plan to pay back m = $800 per month, every month. Define the sequence {Mn} such that Mn is the money that you owe the bank at the beginning of the nth month.
(a) Carefully argue why
Mn+1 = Mn ╱ 1 + 、_ m ; M1 = L
forms an appropriate mathematical model for the sequence Mn .
(b) Write a Matlab code to determine how much money you owe the
bank after 3 years. Using the given values of L and m, and with r = 3, determine how much money you would still owe after 3 years.
(c) Make sure you keep your Matlab code carefully; you may well have the opportunity of using this in the future, with appropriate values for L, m, r and the number of years, if you end up buying a house!
20. Use the Newton-Raphson method to determine the root of cos x = x to 6 decimal places. (Use and modify given Matlab codes to do this, rather than calculating by hand!)
21. (a) In a whiteboard tutorial question, you determined the Newton- Raphson algorithm for finding the square-root of a positive num- ber a. Generalise your method to find the mth root of a, where m e N = {1, 2, 3, 4, . . . }. Verify that this works if m = 1.
(b) Modify Algorithm 2.2 so that you have a working code to deter-
mine the mth root of any given positive number a.
22. Consider using the Newton-Raphson algorithm to determine the root of f(x) := x1/3 = 0 (we know what the answer is!).
(a) Show that the appropriate recurrent relation is
xn+1 = _2xn .
(b) What happens to the sequence xn if you start with x1 = 0.1?
(You should be able to answer this without having to use Matlab.)
(c) Can you figure out why you got this behaviour? Can you guess a value of x1 such that xn converges?
23. An unforced block-spring system with mass m = 1, spring constant k = 13 and damping γ = 6 is governed by the DE
x\\ (t) + 6x\ (t) + 13x(t) = 0 .
Suppose the block-spring system is set into motion with some unknown initial conditions, and you take photographs of it at intervals of 1 time unit, thereby recording the positions x(1), x(2), x(3), etc. From this, you construct a sequence a1 = x(1), a2 = x(2), etc, with the general term being an = x(n). Does the sequence converge, and if so, to what?
dy
dt
(a) Assuming that the stepsize ∆t is fixed, write down the Euler’s
method algorithm for determining numerical solutions to this DE; your equation will tell us how to compute yn+1 in terms of yn .
(b) If yn converges to a finite value , determine a condition that
needs to satisfy. Comment on how this condition is related to the phase-line analysis of Section 1.3.
2022-09-02