Mathematical Foundations

Linear Algebra: Vectors, Matrices, and Tensors

Vectors:

Definition: A vector is an ordered list of numbers (scalars) that represent a point in space or a direction. Vectors can have different dimensions (e.g., 2D, 3D) and are commonly used to represent physical quantities like velocity or force.

Notation: A vector is often written as v or v⃗\vec{v}v, and in component form as [v1,v2,…,vn][v_1, v_2, \dots, v_n][v1​,v2​,…,vn​].

Operations:

• Addition: a⃗+b⃗=[a1+b1,a2+b2,… ]\vec{a} + \vec{b} = [a_1 + b_1, a_2 + b_2, \dots]a+b=[a1​+b1​,a2​+b2​,…]
• Scalar Multiplication: cv⃗=[cv1,cv2,… ]c\vec{v} = [cv_1, cv_2, \dots]cv=[cv1​,cv2​,…]
• Dot Product: a⃗⋅b⃗=a1b1+a2b2+…\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + \dotsa⋅b=a1​b1​+a2​b2​+…
• Cross Product: A vector operation in 3D that produces another vector orthogonal to the two input vectors.

2. Matrices:

Definition: A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent linear transformations, systems of linear equations, and more.

Notation: A matrix is usually written as a capital letter, e.g., A, with elements aija_{ij}aij​ representing the element in the iiith row and jjjth column.

Operations:

• Addition: A+B=[aij+bij]\mathbf{A} + \mathbf{B} = [a_{ij} + b_{ij}]A+B=[aij​+bij​]
• Scalar Multiplication: cA=[caij]c\mathbf{A} = [ca_{ij}]cA=[caij​]
• Matrix Multiplication: A×B\mathbf{A} \times \mathbf{B}A×B involves the dot product of rows and columns.
• Transpose: AT\mathbf{A}^TAT flips the matrix over its diagonal.
• Inverse: A−1\mathbf{A}^{-1}A−1, if it exists, such that AA−1=I\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}AA−1=I (identity matrix).

Tensors:

• Definition: A tensor is a generalization of vectors and matrices to higher dimensions. Tensors are used in deep learning, physics, and more complex data representations.
• Notation: Tensors are often denoted by uppercase letters (e.g., T) with indices representing different dimensions, such as TijkT_{ijk}Tijk​.
• Operations: Tensor operations generalize matrix operations to higher dimensions, including addition, multiplication, and contraction.

Probability and Statistics: Distributions, Bayes’ Theorem

1. Distributions:

Definition: A distribution describes how the values of a random variable are spread or distributed. Common distributions include:

• Normal Distribution: A symmetric, bell-shaped distribution defined by its mean and standard deviation.
• Binomial Distribution: Describes the number of successes in a fixed number of independent Bernoulli trials.
• Poisson Distribution: Describes the number of events occurring within a fixed interval of time or space.

2. Bayes’ Theorem:

Definition: Bayes’ Theorem provides a way to update the probability of a hypothesis based on new evidence. It’s a fundamental theorem in probability theory and statistics.

Formula: P(H∣E)=P(E∣H)⋅P(H)P(E)P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}P(H∣E)=P(E)P(E∣H)⋅P(H)​ where:

• P(H∣E)P(H|E)P(H∣E) is the posterior probability of hypothesis HHH given evidence EEE.
• P(E∣H)P(E|H)P(E∣H) is the likelihood of observing evidence EEE given that HHH is true.
• P(H)P(H)P(H) is the prior probability of HHH.
• P(E)P(E)P(E) is the total probability of observing evidence EEE.

Calculus: Derivatives, Gradients, Optimization

Derivatives:

• Definition: The derivative of a function measures how the function’s output changes as its input changes. It represents the slope of the function at a particular point.
• Notation: The derivative of f(x)f(x)f(x) with respect to xxx is denoted as f′(x)f'(x)f′(x) or df(x)dx\frac{df(x)}{dx}dxdf(x)​.
• Example: For f(x)=x2f(x) = x^2f(x)=x2, the derivative is f′(x)=2xf'(x) = 2xf′(x)=2x.

Gradients:

• Definition: The gradient is a vector of partial derivatives of a multivariable function. It points in the direction of the steepest increase of the function.
• Notation: The gradient of a function f(x,y)f(x, y)f(x,y) is denoted as ∇f\nabla f∇f or grad f\text{grad } fgrad f and is given by [∂f∂x,∂f∂y]\left[ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right][∂x∂f​,∂y∂f​].
• Example: For f(x,y)=x2+y2f(x, y) = x^2 + y^2f(x,y)=x2+y2, the gradient is ∇f=[2x,2y]\nabla f = [2x, 2y]∇f=[2x,2y].

Optimization:

Definition: Optimization involves finding the maximum or minimum value of a function. In calculus, this often involves finding the critical points where the derivative equals zero and determining whether these points are maxima or minima.

Techniques:

• Gradient Descent: An iterative method used to find the minimum of a function by moving in the direction opposite to the gradient.
• Lagrange Multipliers: A method for finding local maxima and minima of a function subject to equality constraints.

Basics of Algorithms and Complexity

1. Algorithms:

Definition: An algorithm is a step-by-step procedure or set of rules to solve a problem or perform a computation. Algorithms are the backbone of computer programming and problem-solving.

Examples:

• Sorting Algorithms: Bubble sort, merge sort, quick sort.
• Search Algorithms: Binary search, depth-first search (DFS), breadth-first search (BFS).

2. Complexity:

Definition: Complexity refers to the computational resources (time and space) that an algorithm requires as the input size grows. It’s often expressed using Big O notation.

Big O Notation:

• O(1): Constant time complexity.
• O(n): Linear time complexity.
• O(n^2): Quadratic time complexity.
• O(log n): Logarithmic time complexity.

Problem: Matrix and Vector Operations

This code example will:

1. Create a vector and a matrix.
2. Perform vector addition and scalar multiplication.
3. Perform matrix multiplication.
4. Compute the dot product of two vectors.
5. Find the transpose of a matrix.

Code Example:

import numpy as np

# 1. Create a vector
vector = np.array([1, 2, 3])
print("Vector:", vector)

# 2. Create a matrix
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print("Matrix:\n", matrix)

# 3. Perform vector addition
vector2 = np.array([4, 5, 6])
vector_sum = vector + vector2
print("Vector Addition:", vector_sum)

# 4. Perform scalar multiplication
scalar = 3
scalar_mult = scalar * vector
print("Scalar Multiplication:", scalar_mult)

# 5. Perform matrix multiplication
matrix2 = np.array([[1, 2, 1], [2, 1, 2], [1, 2, 1]])
matrix_mult = np.dot(matrix, matrix2)
print("Matrix Multiplication:\n", matrix_mult)

# 6. Compute dot product of two vectors
dot_product = np.dot(vector, vector2)
print("Dot Product of vectors:", dot_product)

# 7. Find the transpose of a matrix
transpose = np.transpose(matrix)
print("Transpose of Matrix:\n", transpose)

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