Vectors: Magnitude, Direction, and Properties

Chapter 1: Vectors

• A scalar has only magnitude, examples include temperature, height, and speed.
• A vector has both magnitude and direction in space, its examples are force and velocity.
• Textbooks use bold font for vectors.
• When handwriting, indicate a vector with an arrow symbol over the variable $\overrightarrow{d}$.
• The magnitude of a vector is represented by the absolute value, for example |d|.

Properties of Vectors

• Geometric vectors can be represented with a directed line segment.
• Equal vectors share the same magnitude and direction.
• Equivalent vectors have the same magnitude and direction but may exist at different positions.
• The negative of a vector has the same magnitude but the opposite direction.
• Multiplying a vector by a scalar adjusts its magnitude, and a negative scalar reverses its direction. For example, $2\overrightarrow{v}$ is twice as long as and points the same way as $\overrightarrow{v}$, but $-2\overrightarrow{v}$ is twice as long and points the opposite way.
• A zero vector has a magnitude of zero.

Vector Addition

• Vectors can be added via the triangle or parallelogram law.
• Triangle Law: The resultant vector goes from the tail of the first vector to the head of the second, when the second vector's tail is placed at the first's head.
• Parallelogram Law: The resultant vector goes from the tails of both original vectors to the opposite corner of the completed parallelogram.
• Vector addition is commutative: $\overrightarrow{u} + \overrightarrow{v} = \overrightarrow{v} + \overrightarrow{u}$.
• Vector addition is associative: $(\overrightarrow{u} + \overrightarrow{v}) + \overrightarrow{w} = \overrightarrow{u} + (\overrightarrow{v} + \overrightarrow{w})$.

Vector Subtraction and Scalar Multiplication

• Subtracting a vector is the same as adding its negative: $\overrightarrow{u} - \overrightarrow{v} = \overrightarrow{u} + (-\overrightarrow{v})$.
• Multiplying a vector by a scalar scales the vector's magnitude, and a negative scalar flips its direction.

Unit Vectors

• Unit vectors possess a magnitude of 1.
• A unit vector $\hat{v}$ in the direction of $\overrightarrow{v}$ is equal to $\frac{\overrightarrow{v}}{|\overrightarrow{v}|}$.

Capítulo 4 Espacios Vectoriales

• A vector space consists of a non-empty set $V$ of vectors with defined addition and scalar multiplication operations.

Axioms of Vector Space Addition

• Closure under Addition: For all $\mathbf{u}, \mathbf{v} \in V$, $\mathbf{u} + \mathbf{v} \in V$.
• Commutativity: For all $\mathbf{u}, \mathbf{v} \in V$, $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.
• Associativity: For all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$, $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$.
• Existence of Zero Vector: There exists $\mathbf{0} \in V$ such that for all $\mathbf{u} \in V$, $\mathbf{u} + \mathbf{0} = \mathbf{u}$.
• Existence of Additive Inverse: For every $\mathbf{u} \in V$, there exists $-\mathbf{u} \in V$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$.

Axioms of Scalar Multiplication

• Closure under Scalar Multiplication: For all $\mathbf{u} \in V$ and $c \in \mathbb{R}$, $c\mathbf{u} \in V$.
• Scalar Distributivity: For all $\mathbf{u}, \mathbf{v} \in V$ and $c \in \mathbb{R}$, $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$.
• Vector Distributivity: For all $\mathbf{u} \in V$ and $c, d \in \mathbb{R}$, $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$.
• Scalar Associativity: For all $\mathbf{u} \in V$ and $c, d \in \mathbb{R}$, $c(d\mathbf{u}) = (cd)\mathbf{u}$.
• Scalar Identity Element: For all $\mathbf{u} \in V$, $1\mathbf{u} = \mathbf{u}$.

Linear Combination

• A linear combination of vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ is $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_n\mathbf{v}_n$, where $c_1, c_2, \dots, c_n$ are scalars.

Span

• Given a set of vectors $S = {\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n}$ in a vector space $V$, the span of S, denoted as $\text{span}(S)$, includes all possible linear combinations of vectors from S.

Subspace

• A subset $W$ of a vector space $V$ is a subspace if $W$ is also a vector space under the same operations defined in $V$.
• A non-empty subset $W$ of a vector space $V$ is a subspace if and only if for all $\mathbf{u}, \mathbf{v} \in W$, $\mathbf{u} + \mathbf{v} \in W$ and for all $\mathbf{u} \in W$ and $c \in \mathbb{R}$, $c\mathbf{u} \in W$.

Examples: Subspaces of $\mathbb{R}^2$

• The sets $W_1 = {(x, y) \in \mathbb{R}^2 \mid y = 2x}$, $W_2 = {(x, y) \in \mathbb{R}^2 \mid x = 0}$, and $W_3 = {(x, y) \in \mathbb{R}^2 \mid x + y = 0}$ are subspaces of $\mathbb{R}^2$.

Examples: Non-Subspaces of $\mathbb{R}^2$

• The sets $W_4 = {(x, y) \in \mathbb{R}^2 \mid y = x + 1}$, $W_5 = {(x, y) \in \mathbb{R}^2 \mid y = x^2}$, and $W_6 = {(x, y) \in \mathbb{R}^2 \mid xy = 0}$ are not subspaces of $\mathbb{R}^2$.

Examples: Subspaces of $M_{2 \times 2}(\mathbb{R})$

• The set of all $2 \times 2$ diagonal matrices and the set of all $2 \times 2$ symmetric matrices are subspaces of $M_{2 \times 2}(\mathbb{R})$.

Examples: Subspaces of Polynomials

• The set of all polynomials of degree less than or equal to $m$, where $m < n$, and the set of all polynomials $p(x)$ such that $p(0) = 0$ are subspaces of $P_n(\mathbb{R})$.

Examples: Subspaces of Real Functions

• The set of all even functions and the set of all odd functions are subspaces of $F(\mathbb{R})$.

Quantum Mechanics

• Quantum mechanics describes microscopic object movement.
• Quantum mechanics operates using probabilities rather than certainties.
• Quantum mechanics is valid for macroscopic objects where the quantum effects are negligible.
• Classical mechanics fails to accurately predict the motion of electrons and atoms.
• Classical mechanics is based on knowing the initial conditions to predict a particle's trajectory.

Blackbody Radiation

• Hot objects emit light, the color and intensity of which depend on temperature.
• Classical Physics predicts emitted light should increase indefinitely as wavelength decreases (UV catastrophe).
• Intensity reaches a max and then decreases.
• Max Planck (1900), explained that energy is emitted in quanta, following the formula $E = h\nu$ where h = $6.626 \times 10^{-34} Js$ and $\nu$ is frequency.

Photoelectric Effect

• Shining light on metal can eject electrons.
• Classical physics suggests that the kinetic energy of ejected electrons increases with the intensity of light.
• Albert Einstein(1905) described light as particles called photons: $E = h\nu$, where $KE = h\nu - \phi$; $\phi$ represents the work function.

Atomic Spectra

• Excited atoms emit light at specific wavelengths.
• Niels Bohr (1913) proposed electrons occupy distinct orbits: $E_n = -\frac{R_H}{n^2}$ where $R_H$ = $2.18 \times 10^{-18} J$ and n = 1, 2, 3...
• Classical physics states atoms should emit a continuous spectrum of light instead.

Wave-Particle Duality

• Wave-particle duality dictates that light and particles can behave as both waves and particles.
• Louis de Broglie (1924) described particle wavelength as $\lambda = \frac{h}{p} = \frac{h}{mv}$.

Key Concepts in Quantum Mechanics Include

• Quantization of energy.
• Wave-particle duality.
• The uncertainty principle.
• The uncertainty principle states $\Delta x \Delta p \geq \frac{\hbar}{2}$.

The Poisson Process Definition

• A stochastic process that counts the occurrence of events and their timing within an interval.
• $N(t)$ represents the number of events in the interval $[0, t]$.
• Necessary conditions for a poisson process with $\lambda > 0$: $N(0) = 0$.
• Events in disjoint intervals must be independent
• Events in an interval of length $t$ follows the Poisson distribution: with a mean rate of $\lambda t$.
• Formula: $P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}$, where $n=0,1,2,...$

Interarrival Times

• Each $T_i$ is independent and exponentially distributed with parameter $\lambda$
• $P(T_i > t) = P(N(t) = 0) = e^{-\lambda t}$.
• The probability density function (PDF) of $T_i$ is: $f_{T_i}(t) = \lambda e^{-\lambda t}$, for $t>0$.

Waiting Times

• $S_n = \sum_{i=1}^n T_i$ stands for the time of the nth event.
• The PDF of $S_n$ is: $f_{S_n}(t) = \frac{\lambda^n t^{n-1} e^{-\lambda t}}{(n-1)!}$, for $t>0$, which is a Gamma distribution.

Compound Poisson Processes

• ${N(t), t \geq 0}$ is a Poisson process with rate $\lambda$ in which all ${Y_i, i \geq 1}$ are a family of independent and identically distributed random variables
• These are independent from ${N(t), t \geq 0}$ as well.
• $X(t) = \sum_{i=1}^{N(t)} Y_i$.
• If $E[Y_i] = \mu$ and $Var(Y_i) = \sigma^2 < \infty$, then $E[X(t)] = E[N(t)]E[Y_1] = \lambda t \mu$ and $Var[X(t)] = E[N(t)]E[Y_1^2] = \lambda t (\sigma^2 + \mu^2)$.