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2 43 Vector Operations Based on experience with modeling physical problems, we have an understanding and appreciation of where vectors come from and why we should represent and manipulate them mathematically. The typical operations of vector addition and scalar multiplication, as well as the various vector multiplications, have their origin in well-known physical settings. To a large extent, similar interpretations exist in matrix algebra, which we now address. Given a vector v and a scalar α, an element in the set of real numbers R or complex numbers C, we define αv, the scalar product of α and v, as the vector that has magnitude |α| ||v|| and is otherwise parallel to v.

2004 0. 0 0. 00 018 . 00 0 0. 12: Contours of the absolute error between f and S10 . (a) f(x) = 1, defined in the interval (0, 2), and N = 8. (b) f(x) = 1 if 0 < x < 1, 2 if 1 ≤ x < 3, and N = 16. (c) f(x) = x2 defined in the interval (0, 2), and N = 32. 5. Compute the Fourier sine series of the following functions. In each case plot the contours of the function f and its specified partial sum. (a) f(x, y) = xy(1 − x)(2 − y), in the domain (0, 1) × (0, 2), and N = 8. (b) f(x, y) = sin(x2 + y2 ), in the domain (0, 1) × (0, 1), and N = 32.

A second example of usage of for ... end arises in plotting and displaying several graphs. To illustrate, consider the functions fn (x) = Jn (x), where Jn is the n-th Bessel function of the first kind, which is labeled besselj(n,x) in MATLAB. 7. end command by combining it with the getframe command. 1) end pause(5) end As a final example, let’s consider how for ... end could be used to find a fixed point of function g, a point a that remains fixed under the action of g, that is, g(a) = a. } such that xn converges to a.