Math Notes

Lesson Objectives ▼ Explain what a formal system is: a language, axioms, and rules of inference. Describe a proof as a finite, mechanically checkable sequence of formulas. State the three properties Hilbert wanted: consistency, completeness, and decidability. Understand why the paradoxes of the early 1900s motivated Hilbert's program. Preview what Gödel proved and which of Hilbert's goals it broke. Lesson Outline ▼ A Crisis in the Foundations F What Is a Formal System? Hilbert's Program The Three Properties What Gödel Showed Examples Exercises A Crisis in the Foundations By 1900, mathematics looked like a triumph. Calculus had been made rigorous, geometry had been axiomatized, and the new theory of sets seemed able to describe every mathematical object. Then the foundations cracked. In 1901, Bertrand Russell considered the collection of ...

Lesson Objectives ▼ Define orthogonal vectors and orthonormal sets using the inner product. Understand the orthogonal complement of a subspace. Derive the formula for projecting a vector onto a line. Generalize projection to an arbitrary subspace using the projection matrix. See how orthogonal projection is exactly what the least squares method computes. Lesson Outline ▼ Orthogonal Vectors and Sets Orthogonal Complement Projection onto a Line Projection onto a Subspace Connection to Least Squares Examples Exercises Orthogonal Vectors and Sets Orthogonality Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal when their inner product is zero: \[ \langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u}^T \mathbf{v} = 0. \] Geometrically, this means the two vectors meet at a right angle. A set of vectors is orthogonal when this holds for ev...

Lesson Objectives ▼ Understand the rank of a matrix and its connection to linear independence. Define the null space (kernel) of a matrix and compute it. Identify all four fundamental subspaces of a matrix: column space, null space, row space, and left null space. Apply the Rank-Nullity Theorem to relate rank and nullity. See how these subspaces reveal the structure of a linear transformation. Lesson Outline ▼ Rank of a Matrix Null Space (Kernel) The Four Fundamental Subspaces The Rank-Nullity Theorem Examples Exercises Rank of a Matrix The rank of a matrix \( A \) is the dimension of its column space \( C(A) \) — equivalently, the maximum number of linearly independent columns (or rows) in \( A \). In practice, you find it by row-reducing \( A \) to echelon form and counting the pivot columns: \[ \text{rank}(A) = \text{number of pivots in the row eche...

Lesson Objectives ▼ Understand matrix-vector multiplication through real-world examples. Explore the concept of linearity using grocery shopping data. Connect mathematical operations to practical decision-making. Build intuition for matrix transformations and scaling effects. Prepare for deeper study of matrix algebra and applications. Lesson Outline ▼ Grocery Shopping Example Word Problems Exercises Answers Grocery Shopping Example Imagine a grocery store with 3 items: Bread: $2.50 Milk: $3.20 Apples: $1.80 Five shoppers purchase different quantities: Shopper Bread Milk Apples Alice 2 1 4 Ben 1 2 3 Clara 3 0 2 David 2 3 1 Emma 1 1 5 Matrix-vector multiplication gives us: \[ \begin{bmatrix} 2 & 1 & 4 \\ 1 ...

Lesson Objectives ▼ Understand the concept of matrix inverses and their geometric meaning. Learn conditions for matrix invertibility using determinants. Compute inverses of 2×2 and 3×3 matrices using formulas and row operations. Apply matrix inverses to solve systems of linear equations. Explore applications in data science and machine learning. Lesson Outline ▼ Definition of Matrix Inverse Conditions for Invertibility Computing Matrix Inverses Applications in Data Science Examples Exercises Definition of Matrix Inverse The inverse of a square matrix \( A \) is a matrix \( A^{-1} \) such that: \[ A A^{-1} = A^{-1} A = I \] where \( I \) is the identity matrix. This means applying \( A \) followed by \( A^{-1} \) returns the original vector. Conditions for Invertibility A matrix is invertible if and only if its determinant is non-zero: \[ \text{If...

Lesson Objectives ▼ Understand the concept of vector functions and their derivatives. Learn about partial derivatives and the gradient of a function. Compute divergence and curl of a vector field. Evaluate multiple integrals (double and triple integrals). Apply line and surface integrals. Understand the fundamental theorems of vector calculus. Lesson Outline ▼ Definition of Vector Functions Partial Derivatives and Gradient Divergence and Curl Multiple Integrals: Double and Triple Integrals Line and Surface Integrals Fundamental Theorems of Vector Calculus Examples Definition of Vector Functions A **vector function** assigns a vector to each point in space. It is written as: \[ \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k} \] where \( x(t), y(t), z(t) \) are functions of \( t \). Partial Derivatives and Gradient If ...

Lesson Objectives ▼ Understand the concept of parametric equations and their applications. Convert between parametric and Cartesian equations. Learn the fundamentals of polar coordinates. Convert between polar and Cartesian coordinates. Perform calculus operations on parametric and polar equations. Lesson Outline ▼ Definition of Parametric Equations Eliminating the Parameter Introduction to Polar Coordinates Conversion between Polar and Cartesian Coordinates Calculus with Parametric and Polar Equations Examples Definition of Parametric Equations A curve in the plane can be represented by a set of **parametric equations**: \[ x = f(t), \quad y = g(t), \quad t \text{ is the parameter} \] For example, the parametric equations: \[ x = \cos t, \quad y = \sin t, \quad 0 \leq t \leq 2\pi \] represent a unit circle. Eliminating the Parameter To convert from ...