Vector Space Definition

Definition of a Vector Space

• A vector space over a field $\mathbb{K}$ is a set $E$ with two operations: vector addition ($E \times E \rightarrow E$) and scalar multiplication ($\mathbb{K} \times E \rightarrow E$).
• Vector addition is associative: $\forall u, v, w \in E, (u + v) + w = u + (v + w)$.
• Vector addition is commutative: $\forall u, v \in E, u + v = v + u$.
• There exists an additive identity element $0_E \in E$ such that $\forall u \in E, u + 0_E = u$.
• For each $u \in E$, there exists an additive inverse $-u \in E$ such that $u + (-u) = 0_E$.
• Scalar multiplication is distributive over vector addition: $\forall \lambda \in \mathbb{K}, \forall u, v \in E, \lambda(u + v) = \lambda u + \lambda v$.
• Scalar multiplication is distributive over field addition: $\forall \lambda, \mu \in \mathbb{K}, \forall u \in E, (\lambda + \mu)u = \lambda u + \mu u$.
• Scalar multiplication is associative: $\forall \lambda, \mu \in \mathbb{K}, \forall u \in E, \lambda(\mu u) = (\lambda \mu)u$.
• There exists a multiplicative identity element $1_{\mathbb{K}}$ such that $\forall u \in E, 1_{\mathbb{K}} u = u$.

Examples of Vector Spaces

• $\mathbb{K}^n$ is the set of n-tuples with elements from $\mathbb{K}$.
• $\mathcal{M}_{m,n}(\mathbb{K})$ is the set of $m \times n$ matrices with coefficients in $\mathbb{K}$.
• $\mathbb{K}[X]$ is the set of polynomials with coefficients in $\mathbb{K}$.
• $\mathcal{F}(X, \mathbb{K})$ is the set of functions from $X$ to $\mathbb{K}$.

Subspaces

• A subset $F$ of vector space $E$ is a subspace if it's non-empty, closed under vector addition, and closed under scalar multiplication.
• $F$ is a subspace if $0_E \in F$ and $\forall u, v \in F, \forall \lambda \in \mathbb{K}, \lambda u + v \in F$.

Linear Combinations

• A linear combination of vectors $u_1, u_2,..., u_n$ in $E$ is a vector of the form $\lambda_1 u_1 + \lambda_2 u_2 +... + \lambda_n u_n$, where $\lambda_i \in \mathbb{K}$.

Linear Span (or Vector Space Generated)

• Denoted as $\text{Vect}(S)$ for $S \subseteq E$, it's the set of all possible linear combinations of vectors from $S$.
• Is the smallest subspace of $E$ containing $S$.

Linear Independence, Spanning Sets, and Bases

• A family of vectors $(u_1, u_2,..., u_n)$ is linearly independent if $\lambda_1 u_1 + \lambda_2 u_2 +... + \lambda_n u_n = 0_E \Rightarrow \lambda_1 = \lambda_2 =... = \lambda_n = 0$.
• A family of vectors $(u_1, u_2,..., u_n)$ spans $E$ if $E = \text{Vect}(u_1, u_2,..., u_n)$.
• A family of vectors $(u_1, u_2,..., u_n)$ is a basis of $E$ if it is both linearly independent and spans $E$.

Dimension of a Vector Space

• If $E$ has a finite basis, all bases of $E$ have the same number of elements.
• This number is the dimension of $E$, denoted as $\dim(E)$.

Sum of Subspaces

• If $F$ and $G$ are subspaces of $E$, their sum is $F + G = {u + v \mid u \in F, v \in G}$.
• $F + G$ is a subspace of $E$.

Direct Sum

• The sum $F + G$ is called a direct sum, denoted $F \oplus G$, if every vector in $F + G$ can be uniquely written as the sum of a vector from $F$ and a vector from $G$.
• $F + G$ is a direct sum if and only if $F \cap G = {0_E}$.
• If $F \oplus G = E$, then $F$ and $G$ are supplementary in $E$.

Rank Theorem

• For a linear map $f : E \rightarrow F$ between finite-dimensional vector spaces, $\dim(E) = \dim(\text{Ker}(f)) + \dim(\text{Im}(f))$, where $\text{Ker}(f)$ is the kernel and $\text{Im}(f)$ is the image of $f$.

NetApp ONTAP CLI

Introduction

• This quick start guide provides an introduction to the NetApp ONTAP Command Line Interface (CLI). The CLI is a powerful tool to manage and configure storage systems.

Prerequisites

• A running NetApp storage system is required.
• Network access to the system.
• An SSH client installed on the computer.
• Basic storage concepts knowledge is a must

Accessing the ONTAP CLI

• Use an SSH client to connect to the IP address of the storage system.
• Open the SSH client, input the IP address of your system as the "Hostname". Ensure "Connection type" is "SSH" and click "Open". Enter username and password when prompted.

Basic commands

• help: Displays available commands.
• man: Shows the manual page for a specified command.
• version: Displays the ONTAP version running.
• date: Shows current system date and time.
• uptime: Displays how long the system has been running.
• system status show: Displays basic system status information.

Navigating the CLI

• The CLI is organized in a command hierarchy. Use cd to navigate. For example, cd / changes to the top level of the hierarchy.
• The help command lists available commands at the current level.

Getting Help

• Use man for help with specific commands; for example: man volume create. It shows detailed information including syntax, options, and examples..

Using Scripts with the CLI

• The CLI can automate storage management tasks by creating and running scripts containing a series of commands.
• Run scripts with the cli -f command, where `` is the script filename.

Photoelectric Effect

• Electrons emit from a metal surface when irradiated by light.
• Minimum energy $W$ (work function) is required to release electrons.
• Maximum kinetic energy of photoelectrons: $E_{kin} = h\nu - W = h\frac{c}{\lambda} - W$, where $\nu$ is frequency and $\lambda$ is wavelength.

Experimental data

• Experimental data are provided for three metals with cutoff wavelengths:
  • Sodium: $\lambda_{cutoff} = 680 nm$
  • Potassium: $\lambda_{cutoff} = 560 nm$
  • Cesium: $\lambda_{cutoff} = 650 nm$

Tasks

• Calculate work functions for each metal in electron volts (eV), given that $1 eV = 1.602 * 10^{-19} J$.
• Describe how the kinetic energy of the photoelectrons changes when light intensity is doubled or when the light's wavelength is halved.

Compton Effect

• A photon transfers energy to a particle during scattering.
• Determine the increase in a photon's wavelength by a factor of ten after scattering, given initial wavelengths of:
  • a) $1 \mu m$
  • b) $500 nm$

Bohr Model of the Hydrogen Atom

• An electron orbits an atomic nucleus (electron and proton) in a circular path, with quantized angular momentum $L = n\hbar, n \in \mathbb{N}$.
• Coulomb force and centripetal force determine the electron's radius.

Tasks

• Determine an expression for the radius $r_n$ in terms of velocity $v_n$ using the Coulomb force and centripetal force.
• Determine an expression for velocity $v_n$ in terms of $n$ using the Bohr postulate (quantization of angular momentum).
• Determine $r_n$ as a function of $n$, and calculate the Bohr radius $a_0$ for $n = 1$.

Uncertainty Principle

• Use the uncertainty principle to demonstrate that electrons cannot exist within the atomic nucleus.
• The diameter of an atomic nucleus is approximately $10^{-14}m$.

Tasks

• Calculate the minimum momentum of electrons if they were located within the nucleus.
• Calculate the associated velocity and kinetic energy and interpret the result.
• The rest energy of an electron is given as $0.51 MeV = 8.187 * 10^{-14} J$.

Einstein Field Equation Review

• Covariant derivatives of tensors transform as tensors and are used to define the curvature tensor $R^{\mu}_{\nu \rho \sigma}$.

The Einstein Field Equation

• A relativistic generalization of Poisson's equation is needed to describe how matter curves spacetime: $G_{\mu\nu} = \kappa T_{\mu\nu}$.

Conditions on $G_{\mu\nu}$

• It must be a tensor.
• It must depend only on the metric and its first and second derivatives.
• It must be symmetric.
• It must be conserved, following $\nabla_{\mu} G^{\mu\nu} = 0$.

General Tensor Formula

• The most general tensor meeting conditions 1 and 2 is $G_{\mu\nu} = c_1 R_{\mu\nu} + c_2 R g_{\mu\nu} + c_3 \Lambda g_{\mu\nu}$.
• $R_{\mu\nu} \equiv R^{\alpha}{\mu \alpha \nu}$ is the Ricci tensor, $R \equiv R^{\mu}{\mu}$ is the Ricci scalar, $\Lambda$ is the cosmological constant, and $c_i$'s are constants.

Applying the Bianchi Identity

• Contracting the Bianchi identity $\nabla_{\lambda} R_{\mu\nu\rho\sigma} + \nabla_{\rho} R_{\mu\nu\sigma\lambda} + \nabla_{\sigma} R_{\mu\nu\lambda\rho} = 0$ gives $\nabla_{\mu} R^{\mu\nu} = \frac{1}{2} \nabla^{\nu} R$.

Einstein's Choice

• Einstein's equation sets $c_1 = 1$ and enforces the conservation condition, resulting in $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu}$.
• This gives the Einstein field equation: $R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}$.

Definition of the Natural Logarithm

• The natural logarithm function, $ln$, is defined on $]0; +\infty[$ as the primitive of $x \longmapsto \frac{1}{x}$ that equals zero at $x=1$.
• Key Values: $ln(1) = 0$, $ln(e) = 1$.
• Inverse Relationships: For $x > 0$, $e^{ln(x)} = x$, and for all real x, $ln(e^x) = x$.

Algebraic Properties

• Logarithm of Product: $ln(ab) = ln(a) + ln(b)$
• Logarithm of Reciprocal: $ln(\frac{1}{a}) = -ln(a)$
• Logarithm of Quotient: $ln(\frac{a}{b}) = ln(a) - ln(b)$
• Logarithm of Power: $ln(a^n) = n \cdot ln(a)$
• Logarithm of Square Root: $ln(\sqrt{a}) = \frac{1}{2} ln(a)$

Properties of the Natural Logarithm

• $x$ approaches 0, $ln(x)$ approaches negative infinitiy
• $x$ apporaches the positive infinity, $ln(x)$ approaches to the positive infinity
• $x$ apporaches the positive infinity, a quotient $ln(x)/x$ appoaches 0
• $x$ approaches 0, a multiple $x ln(x)$ approaches 0

Derivative of $ln(x)$

• The derivative of $ln(x)$ is given by $(ln(x))' = \frac{1}{x}$.

Variations

• The function $ln$ is strictly increasing on $]0; +\infty[$.

Representational curve

• Function starts from negative infinity, increasing infinitely and crosses x-axis @ (1,0)

Angular Momentum Definition

• Angular momentum of a particle with respect to a fixed point: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$.
  • $\vec{L}$ is angular momentum, $\vec{r}$ is the position vector, $\vec{p}$ the linear momentum with $m$ as mass and $\vec{v}$ as velocity.

Rotation around a Fixed Axis

• Angular momentum of a rigid body rotating around fixed axis: $L = I\omega$, where $I$ is the moment of inertia and $\omega$ the angular velocity.

Torque and Angular Momentum

• The rate of change of angular momentum equals the applied torque: $\tau = \frac{d\vec{L}}{dt}$.
  • $\tau$ being torque

Conservation of Angular Momentum

• If the external torque is zero, the total angular momentum of the system is conserved: $\vec{L} = \text{constant}$.

Angular and Linear variable similarities

• Position: $x = r\theta$
• Velocity: $v = r\omega$
• Acceleration: $a = r\alpha$
• Magnitude : $\tau = rF$
• Momentum: $L = I\omega = r\cdot p$

DTFT From Fourier Series

• Begin with Fourier series analysis formula: $a_k = \frac{1}{T} \int_{} x(t)e^{-jk\Omega t} dt$, with $\Omega = \frac{2\pi}{T}$.

Approximating $x(t)$

• Sample $x(t)$ using Dirac delta function, representing using $\hat{x}(t)$ $$x[n] = x[nT]$$ $$ \hat{x}(t) = \sum_{n=-\infty}^{\infty} x[n] \delta(t - nT) $$

Deriving DTFT

• Fourier Transform of $\hat{x}(t)$: $$ X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} $$

Inverse DTFT

• The inverse Discrete-Time Fourier Transform (DTFT) is: $$ x[n] = \frac{1}{2\pi} \int_{} X(e^{j\omega}) e^{j\omega n} d\omega $$

Property of the Functions

• DTFT is periodic with period $2\pi$
• DTFT has Continuous frequency on the range $[-\pi, \pi]$

Bernoulli's Principle

• Increase in fluid speed occurs with decrease in pressure or potential energy.

Airplane Flight Application

• Air flows faster over the wing, decreasing pressure and creating lift.

Airfoil Definition

• It is a structure with curved surfaces designed to provide lift when air flows around it

Lift

• It is generated with more pressire @ bottom of wing vs top

Bernoulli's Equation

$$P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$$

Simplified

$$P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2$$

Explore the definition of a vector space over a field, covering vector addition and scalar multiplication properties. Key axioms include associativity, commutativity, and the existence of additive identity and inverse elements. Understand the distributive and associative nature of scalar multiplication.