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Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors 1. What are Eigenvalues and Eigenvectors? When a linear transformation is applied to a vector, some vectors only scale without changing direction. These special vectors are called eigenvectors , and the scaling factor is called the eigenvalue . Mathematical Definition A v = λ v Where A is a square matrix, v is an eigenvector, and λ is the eigenvalue. Real-World Meaning Physics: Eigenvectors represent stable states (e.g., vibration modes in mechanics). Machine Learning: PCA finds important patterns using eigenvectors. Computer Graphics: Used in transformations like shearing and rotations. 2. How to Compute Eigenvalues and Eigenvectors To find the eigenvalues of a matrix A , solve the characteristic equation: det(A - λI) = 0 Steps: Compute A - λI , where I is the identity matrix. Solve det(A - λI) = 0 to find eigenvalues λ . For each λ , solve (A - λ...

Transformations Over Time Transformations Over Time 1. What Does “Over Time” Mean? When we apply a transformation repeatedly or continuously over time, the result is a dynamic system . The transformation evolves based on time ( t ) or steps ( n ). Example: Rotating a shape frame by frame in an animation. Example: A car moving along a curved path using translations and rotations. 2. Linear Transformations Over Time A linear transformation over time follows the equation: v t+1 = A v t where: v t is the vector at time t . A is the transformation matrix. v t+1 is the transformed vector at t+1 . Example: Scaling Over Time A = [ 2 0 ] [ 0 2 ] Applying it repeatedly leads to exponential growth : v1 = A v0, v2 = A v1, v3 = A v2 3. Continuous Transformations (Differential Equations) Some transformations happen smoothly over time and are described by differential equations : d v / dt = ...

Common Transformation Types and Their Formulas Common Transformation Types and Their Formulas 1. Scaling Transformation Scaling changes the size of an object by multiplying each coordinate by a scale factor. In 2D, a scaling transformation can be represented by the following matrix: S = [ [sx, 0], [0, sy] ] Where sx and sy are the scaling factors for the x and y axes, respectively. Real-World Meaning: Scaling is used in image resizing, zooming in/out on maps, or adjusting the size of objects in computer graphics. Algorithm Steps: Multiply the x-coordinate by sx . Multiply the y-coordinate by sy . Reconstruct the vector with the new coordinates. 2. Rotation Transformation Rotation turns an object around a fixed point (usually the origin) by a given angle θ . In 2D, the rotation matrix is: R(θ) = [ [cos(...

Linear Transformations Linear Transformations 1. What is a Linear Transformation? A linear transformation is a function that maps vectors from one vector space to another while preserving vector addition and scalar multiplication. Real-World Meaning: Image Processing: Transforming an image (scaling, rotating, shearing). Computer Graphics: Converting 3D objects into 2D projections. Physics: Changing reference frames in mechanics or relativity. 2. Properties of Linear Transformations A function T: V \to W is a linear transformation if it satisfies: Additivity: T(u + v) = T(u) + T(v) Homogeneity: T(c v) = c T(v) Real-World Meaning: Resizing an image: A transformation that doubles the size of an image maintains proportionality. Rotating a figure: The shape stays the sa...

Vector Spaces with Real-World Context Vector Spaces with Real-World Context 1. Definition of a Vector Space A vector space is a set of vectors that can be added together and multiplied by scalars, following specific rules (axioms). These properties allow vectors to be manipulated in a consistent way. Real-World Example: Geometric Space: Any position or direction in a 2D or 3D world can be represented as a vector. Function Spaces: The set of all polynomials with real coefficients forms a vector space. 2. Properties of a Vector Space (Axioms) Addition Properties: u + v = v + u (Commutativity) (u + v) + w = u + (v + w) (Associativity) v + 0 = v (Zero Vector Exists) v + (-v) = 0 (Additive Inverse Exists) Scalar Multiplication Properties: c (d v) = (cd) v (Associativity of Scalar Multiplication) c (u + v) = c u + c v (Distributivity of Scalar over Vector Add...

Fundamentals of Vectors Fundamentals of Vectors 1. Definition of a Vector A vector is a mathematical object that has both magnitude (size/length) and direction . They allow us to describe relationships and movements in ways scalars cannot. Real-World Examples: Velocity: Speed with direction (e.g., 60 km/h east). Force: A push or pull applied in a direction. Electric Fields: Represented as vectors at every point in space. 2. Notation of Vectors Vectors can be represented in different forms: Column Form: \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} Unit Vector Notation: \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} Graphical Representation: Drawn as an arrow in a coordinate plane. 3. Types of Vectors ...

Prerequisites for Understanding Vectors (with Real-World Meaning) Prerequisites for Understanding Vectors 1. Numbers and Arithmetic Natural Numbers (ℕ): Counting numbers (1, 2, 3, ...). Real-World Meaning: Used in everyday counting—like the number of books on a shelf or apples in a basket. Integers (ℤ): Includes negative numbers (-2, -1, 0, 1, 2, ...). Real-World Meaning: Represent situations like temperatures (below zero), financial debts, or elevations above and below sea level. Rational Numbers (ℚ): Fractions and decimals. Real-World Meaning: Used in measurements and everyday situations like recipes or sharing a pizza equally. Real Numbers (ℝ): Continuous values including π and √2. Real-World Meaning:...