# Sample document using L TEX A Cornell T. Moose ```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 &times; 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 &gt; 0}.)
m
Let D ⊆ X. Recall that a function f : D → Y between Banach spaces has
a limit L at x0 ∈ X if ∀ &gt; 0, ∃δ such that |x − x0 | &lt; δ ⇒ |f (x) − L| &lt; . 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 &times; 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
```