1. [11 marks] This problem addresses the problem how to find the representation of a vector (either real-space or from a function space) with respect to a set of given orthogonal basis vectors. Part a) to d) are establishing the principle for the intuitive case of R³, and part e) is asking you to perform the same task for an co-dimensional function space. (a) Consider the basis for R³ consisting of the vectors r1 = (1,1,0), r2 = (1,-1,0) and r3 = (0,0,2). Convince yourself that these vectors are pairwise orthogonal and linearly indepen- dent. (b) Express the vector f = (13,8,7 as the linear combination f = C2-1 Akrk. Determine the coeffi- cients Ak by solving the relevant system of linear equations. (c) Now perform the same task, but by using orthogonality of the basis vectors instead of solving the linear system of equations. Convince yourself that you obtain the result. (d) Why does the 'orthogonality method' above always work for orthogonal bases? To show this, consider a vector space whose basis consists of n orthogonal vectors V1, V2, Vn. Then ex- press a given vector W as Ex=1 n BkVk (Why can you be certain that this is possible?). Now show that Bk = (e) Consider an infinite set of continuous functions P1(x),42(x),03(x) defined for X E [0,1] and assume that they are mutually orthogonal with respect to the inner product (Pk,Qi) = Jo 1 Ok (x) Qj(x)dx: = for k # j (1) 3 fork =j. 1 Use orthogonality to find the coefficients Ck (k = 1,2,3, x) such that CK=1 CKQK(x) = g(x) for some known function 8(x) defined for X E [0,1]. . 2. [10 marks] Suppose the tide heights in Fremantle are given by H (t) = (2rt 2 1 ) + 1.8 cos (37t) 7 (2) where t is time measured in days. (a) Use a computer package (Matlab, Excel, Scientific Notebook, etc) to plot these functions over a 90 day interval. Does the function appear to be periodic? If so, estimate the period of the function H(t) from the graphical plot. (b) Now, justify without reference to the graphical plot if the function is periodic. If it is periodic, determine the fundamental period. Check that your results are consistent with those of part (a). (c) Discuss if one of the two terms of H(t) can be interpreted as the diurnal (daily) tide. (d) Write this function as a "complex Fourier series" using the Euler equations. We simply want you to replace the cos and sin terms using Euler's equations and reorder by powers of e. 3. [14 marks] Write the Fourier series of the 2-periodic function 2 if 0 X < 1 f(x) = (3) - -2 + x² if 1 < X < 2. Plot and compare your Fourier series with the actual function f (xx, on the same plot. You may look up integrals of the type f x2 sin (Bx), etc in integral tables or using Wolfram Alpha. Hint: Given that this is a 2-periodic function, the usual equations for the coefficients an = 1/2 J L L and bn = apply. Evidently, this function f (x) is piecewise defined, but not broken down into the intervals [ - -1,0) and [0,1) but into the intervals 0, 1) and (1,2). For the solution, consider if the integrals 5°1 2 can be replaced by integrals over 1 and why. 4. [15 marks] Consider the function h(x) = (1 - 1/3 on X € [0,2]. (a) Sketch the even and odd periodic extensions of the function over the interval -2,4]. (b) Write both the Fourier sine and cosine series for this function. (c) Using Matlab or similar, plot the function and both Fourier series using 10 terms of the full interval on the same axes and compare. Comment on whether the convergence of both series is in line with expectation.