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Sample document using LATEX Cornell T. Moose October 2014 Linear algebra is a pivotal tool in 100% of mathematics. A couple applications of linear algebra include: 1. Quantum mechanics 2. Web search engine algorithms An m × n real-valued matrix M can be viewed as a function from Rn to R . In this context, we call M a matrix function. For example, take m = 2 and n = 10. If x ∈ R10 , y = M x is a vector in R2 . Similar statements carry over when working over N, Z, Q, or C. (There is a feudal war over whether N should be {n ∈ Z : n ≥ 0} or {n ∈ Z : n > 0}.) m Let D ⊆ X. Recall that a function f : D → Y between Banach spaces has a limit L at x0 ∈ X if ∀ > 0, ∃δ such that |x − x0 | < δ ⇒ |f (x) − L| < . In this case, we write lim f (x) = L x→x0 A function f is continuous at x0 ∈ X if f is defined at x0 and the limit of f at x0 is f (x0 ). Theorem 0.1. An m × n matrix function M is continuous at all x0 ∈ Rn . Proof. Obvious. Sums, integrals, and derivatives Now, we move on to some harder analysis. ∞ X 1 π2 = n2 6 n=1 The statement of (1) is known as Basel’s problem. It is well-known that Z 1 x3 dx = 0 −1 1 (1) Table 1: Trigonometric values theta 0 π/4 π/2 π cos θ √1 2/2 0 -1 sin θ √0 2/2 1 0 Figure 1: This figure shows an example of adaptice mesh refinement (amr). d Note, however, that if g(x) = sin x, then dx g(x) = cos(x). This is also sometimes 9 dg d 0 1 [ln(3(sec x) ) − x2 e log11 x]. denoted dx or g (x). As an exercise , compute dx Tables and figures Sometimes data needs to be displayed in a table or figure. A good example may be trigonometric data. See Table . Figures are entered with the includegraphics command. A figure showing an AMR setup can be seen in Figure . Note: LATEXdecides where to put figures and tables. 1 Hint, use the Chain Rule! 2