Mauricio Poppe notes

Multiplying a vector by a matrix applies one transformation.

As seen in projective geometry , the perspective phenomenon is where an object appears to be smaller the further away it is from the point of view.

Orthographic Projection A projection is a dimension-reducing operation.

2D Translation A translation is an affine transformation, which is a linear transformation followed by some displacement. $$ \mathbf{v^\prime} = \mathbf{Mv} + \mathbf{b} $$ Even though we can’t express 2D translation using a 2x2 matrix, we can express such

Euler angles describe the orientation of a rigid body with three numbers: yaw, pitch, and roll.

2D Shearing In 2D, we can skew points towards the $x$-axis by making $x^\prime = x + sy$.

Rotation transforms orient geometry in 2D and 3D space about an origin or axis without altering size or shape. 2D Rotation A 2D rotation depends on a single parameter, the angle $\theta$.

Scaling Along the Cardinal Axes Intuitively, the basis vectors should be multiplied by a scalar.

A transformation moves geometry: translation shifts it, rotation spins it, scale resizes it.

The camera’s view transform applies the same machinery from the camera’s point of view; see The View Transform .

Imaginary Numbers Invented to solve problems where an equation has no real roots, e.g., $x^2 + 16 = 0$.

A cycle that contains every vertex of a graph $G$ is called a Hamiltonian cycle.

A circuit $C$ in a graph $G$ is called an Eulerian circuit if $C$ contains every edge of $G$ (remember that a circuit is a closed trail, i.e., a walk in which no edge is traversed more than once and that begins and ends at the same vertex).

Unweighted graph We call a shortest path from vertex $u$ to vertex $v$ a path of length $k$, where the path consists of vertices $p = (x_1, x_2, \ldots, x_k)$ such that $x_1 = u$, $x_k = v$, and $k$ is minimum.

A graph $G$ is called acyclic if it has no cycles.

A connected subgraph of $G$ that is not a proper subgraph of any other connected subgraph of $G$ is a component of $G$; i.e., there’s a $u-v$ path in the mentioned subgraph.

If a connected graph $G$ of order $n$ has no cycles, then, of course, $G$ is a tree.

All the facts/properties below are considered for an undirected, connected graph $G$.

Undirected graph In the following undirected graph $G$, the edges $v_2v_3$ and $v_3v_4$ are bridges.

Let $G$ be a digraph.

Breadth First Search (BFS) Given a graph $G$ and a distinguished source vertex $s$, BFS explores the edges of $G$ to discover the vertices adjacent to $s$.

A graph is a pair $G = (V, E)$.

The fundamental theorem of arithmetic states that every positive integer can be written uniquely as a product of primes example: $$ 65340 = 2^2 \cdot 3^3 \cdot 5 \cdot 11^2 $$ Trial division Trial division is the simplest algorithm for factoring an intege…

The divisor function represented as $d(n)$ counts the number of divisors of an integer.

A prime number is a natural number greater than $1$ which has no positive divisors other than $1$ and itself Naive test Let $n$ be the number we want to check for primality.

Let $n!

Given two numbers $n$ and $k$ find the greatest power of $k$ that divides $n!

Let $a$, $b$, and $x$ be positive real numbers such that $$ a^x = b $$ We want to find the value of $x$.

Let $p_1, p_2, \ldots, p_n$ be distinct numbers relatively prime.

Congruence relation For a positive integer $n$, two integers $a$ and $b$ are said to be congruent modulo $n$ if the remainders of $a / n$ and $b / n$ are the same.

Bezout’s identity For non-zero integers $a$ and $b$, let $d$ be the greatest common divisor $d = gcd(a, b)$.

Euclid’s Algorithm for finding the Greatest Common Divisor of two or more integers is based on the following observations: if $x = y$ then $$ gcd(x, y) = gcd(x, x) = x $$ if $x > y$ then $$ gcd(x, y) = gcd(x - y, y) $$ Proof: suppose that $d$ is a divisor…

Definition This algorithm finds prime numbers up to a number $n$.

Algorithm description Finding $a^n$ involves performing $n$ multiplications of $a$.

Examples $$ \begin{align*} \phi(1) &= 1 \quad (1) \\ \phi(2) &= 1 \quad (1) \\ \phi(3) &= 2 \quad (1, 2) \\ \phi(4) &= 2 \quad (1, 3) \\ \phi(5) &= 4 \quad (1, 2, 3, 4) \\ \phi(6) &= 2 \quad (1, 5) \end{align*} $$ Properties The following three properties…

There are simple functions for which we cannot find antiderivatives in terms of the functions we know.

We’re asked to find the derivative of the following function with respect to $x$: $$ \begin{equation}\label{definition-function} y = 3x^2 \end{equation} $$ Performing the differentiation process: $$ \begin{align*} y' &= \lim_{\Delta{x} \to 0} \frac{f(x +…

Physical Interpretation of the Derivative The primary concept of calculus deals with the rate of change of one variable with respect to another.

Why?

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