Vectors in R^n, Linear Combinations

Vectors in $\mathbb{R}^n$

• A vector in $\mathbb{R}^n$ is an ordered list of $n$ real numbers called components: $\vec{v} = (v_1, v_2,..., v_n)$, where $v_i \in \mathbb{R}$ for $i = 1, 2,..., n$.
• Vector addition is performed component-wise: $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2,..., u_n + v_n)$.
• Scalar multiplication involves multiplying each component of the vector by the scalar: $c\vec{v} = (cv_1, cv_2,..., cv_n)$, where $c \in \mathbb{R}$.

Linear Combinations and Independence

• A linear combination of vectors $\vec{v_1}, \vec{v_2},..., \vec{v_k}$ is given by $c_1\vec{v_1} + c_2\vec{v_2} +... + c_k\vec{v_k}$, where $c_i \in \mathbb{R}$.
• Vectors $\vec{v_1}, \vec{v_2},..., \vec{v_k}$ are linearly independent if $c_1\vec{v_1} + c_2\vec{v_2} +... + c_k\vec{v_k} = \vec{0}$ implies $c_1 = c_2 =... = c_k = 0$.

Dot Product and Orthogonality

• The dot product of two vectors $\vec{u}$ and $\vec{v}$ is $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 +... + u_nv_n$.
• Two vectors $\vec{u}$ and $\vec{v}$ are orthogonal if their dot product is zero: $\vec{u} \cdot \vec{v} = 0$.

Vector Norm and Distance

• The norm (or length) of a vector $\vec{v}$ is $||\vec{v}|| = \sqrt{\vec{v} \cdot \vec{v}} = \sqrt{v_1^2 + v_2^2 +... + v_n^2}$.
• The distance between two vectors $\vec{u}$ and $\vec{v}$ is $d(\vec{u}, \vec{v}) = ||\vec{u} - \vec{v}||$.

Matrices

• A matrix is a rectangular array of numbers, with a matrix $A$ of size $m \times n$ having $m$ rows and $n$ columns.

Matrix Operations

• Matrix addition is $A + B = [a_{ij} + b_{ij}]$ (if $A$ and $B$ are of the same size).
• Scalar multiplication is $cA = [ca_{ij}]$.
• Matrix multiplication is $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$ (if the number of columns of $A$ equals the number of rows of $B$).

Matrix Types

• A matrix is square if $m = n$.
• The identity matrix $I_n$ has 1s on the diagonal and 0s elsewhere.
• The transpose of a matrix $A^T$ has elements $(A^T){ij} = a{ji}$.
• The inverse of a matrix $A^{-1}$ satisfies $AA^{-1} = A^{-1}A = I$.

Determinant of a Matrix

• The determinant of a square matrix $A$, denoted det(A) or |A|, is a scalar value.
• For a $2 \times 2$ matrix: $\begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc$.
• For a $3 \times 3$ matrix, the rule of Sarrus or Laplace expansion can be used.

Linear Equation Systems

• Systems of linear equations can be written in the form $Ax = b$, where $A$ is the coefficient matrix, $x$ is the vector of unknowns, and $b$ is the constant vector.

Methods for Solving Linear Systems

• Gauss Elimination
• Cramer's Rule (if det(A) $\neq$ 0)
• Matrix Inversion (if A is invertible: $x = A^{-1}b$)

Vector Spaces

• A vector space $V$ is a non-empty set with addition and scalar multiplication operations that satisfy certain axioms.

Examples of Vector Spaces

• $\mathbb{R}^n$
• The set of $m \times n$ matrices
• The set of continuous functions on a given interval

Vector Subspaces

• A subset $W$ of a vector space $V$ is a subspace if $W$ is itself a vector space with the same operations as $V$.
• A subset $W$ of $V$ is a subspace if: it contains the zero vector, it is closed under addition, and it is closed under scalar multiplication.

Vector Space Basis and Dimension

• A basis of a vector space $V$ is a set of linearly independent vectors that span $V$.
• The dimension of a vector space $V$, denoted dim(V), is the number of vectors in a basis of $V$.

Linear Transformations

• A linear transformation $T: V \rightarrow W$ is a function preserving addition and scalar multiplication: $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$ and $T(c\vec{v}) = cT(\vec{v})$.

Eigenvalues and Eigenvectors

• An eigenvector of a square matrix $A$ is a non-zero vector $\vec{v}$ such that $A\vec{v} = \lambda\vec{v}$ for a scalar $\lambda$, called the eigenvalue.

Calculating Eigenvalues and Eigenvectors

• Find eigenvalues by solving the characteristic equation: det$(A - \lambda I) = 0$.
• For each eigenvalue $\lambda$, find eigenvectors by solving the system $(A - \lambda I)\vec{v} = \vec{0}$.

Diagonalization

• A matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$.
• The columns of $P$ are the eigenvectors of $A$, and the diagonal elements of $D$ are the corresponding eigenvalues.

Bernoulli's Principle

• For an inviscid flow, an increase in fluid speed occurs simultaneously with a decrease in pressure or potential energy.

How Wings Generate Lift

• Wings are shaped to make air flow faster over the top, resulting in low pressure, and slower under the bottom, resulting in high pressure; high pressure under the wing pushes it upwards.

Bernoulli's Equation

• Mathematical representation: $P + \frac{1}{2} \rho v^2 + \rho g h = constant$, where $P$ is static pressure, $\rho$ is fluid density, $v$ is fluid speed, $g$ is gravity, and $h$ is elevation.

Simplified Bernoulli's Equation

• When elevation differences are negligible: $P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2$; as speed increases, pressure decreases.

Venturi Effect

• A reduction in fluid pressure results when a fluid flows through a constricted section of a pipe.

Applications of Venturi Effect

• Uses: Carburetors, spray bottles, pipelines, and chimneys.

Remote Sensing

• Remote sensing involves acquiring information about objects or phenomena without physical contact.

Applications of Remote Sensing

• It is used in agriculture, forestry, urban planning, natural disaster management, and climate studies.

Remote Sensing Platforms

• Platforms are classified by altitude: spatial (satellites, space stations), airborne (airplanes, helicopters, drones), and terrestrial (vehicles, tripods, towers).

Remote Sensing Sensors

• Passive sensors measure emitted or reflected electromagnetic radiation: cameras, radiometers, and spectrometers.
• Active sensors emit their own radiation and measure backscatter: radars and lidars.

Remote Sensing Data Processing

• Raw data processing includes geometric and atmospheric correction, classification, and analysis.

Applications of Remote Sensing

• Used in mapping land cover, monitoring deforestation, tracking crops, assessing natural disaster damage, and studying pollution.

Advantages of Remote Sensing

• Offers broad spatial and temporal coverage, relatively low cost, and access to inaccessible areas.

Disadvantages of Remote Sensing

• Requires complex data processing, sensitive to atmospheric conditions, and can have limited spatial resolution.

Lab 3: Equilibrium - Objectives

• Use static equilibrium to determine unknown forces.
• Resolve forces into x and y components.
• Apply conditions for static equilibrium to solve force problems.

Lab 3: Equilibrium - Introduction

• An object is in equilibrium when the sum of forces equals zero, indicating it is at rest or moving at a constant velocity.
• Static equilibrium requires that the object is at rest.

Conditions for Static Equilibrium

• The vector sum of forces must equal zero.
• The net torque acting on the object must equal zero.
• In a plane, this means the sum of x-components and y-components of forces must equal zero.

Lab Procedure (Part 1: Equilibrium of a Particle)

• Use a force table to study a particle's equilibrium by applying forces via strings and masses.
• Adjust masses and angles until the ring on the table is centered.
• Record masses and angles, calculate x and y components of each force, and confirm their sums are close to zero in equilibrium.

Lab Procedure (Part 2: Equilibrium of a Rigid Body)

• Use a meter stick suspended by two spring scales to study rigid body equilibrium.
• Hang masses from the stick and record spring scale readings.
• Calculate torques due to each force and confirm that their sum is close to zero in equilibrium.

Lab Report Requirements

• Include introduction, equipment description, procedure, data, results, discussion of errors, and answers to questions.

Sample Lab Questions

• What are the conditions for static equilibrium?
• How can static equilibrium be used to find unknown forces?
• What potential errors exist in this experiment?

Algorithmic Complexity - Definition

• Algorithmic complexity measures the resources (time, memory) an algorithm needs to solve a problem of a given size, used to quantify and compare algorithm efficiency.

Types of Complexity

• Time complexity measures the execution time as a function of input size.
• Space complexity measures the memory space needed as a function of input size.

Asymptotic Notation

• Asymptotic notation describes the limiting behavior of complexity as input size approaches infinity.

Common Asymptotic Notations

• Big O Notation (O) represents the upper bound or worst-case scenario.
• Omega Notation ($\Omega$) represents the lower bound or best-case scenario.
• Theta Notation ($\Theta$) represents the average-case complexity.

Common Complexity Classes

• Constant Time - O(1): Execution time is constant regardless of input size.
• Logarithmic Time - O(log n): Execution time increases logarithmically with input size.
• Linear Time - O(n): Execution time increases linearly with input size.
• Log-Linear Time - O(n log n): Execution time increases proportionally to n multiplied by the logarithm of n.
• Quadratic Time - O(n^2): Execution time increases quadratically with input size.
• Exponential Time - O(2^n): Execution time increases exponentially with input size.