A Gentle Introduction to Mathematics for Machine Learning

Discover how machine learning uses mathematics to turn data into predictions, modeling y hat as f(x) and training parameters to approximate the exact function y with inputs x1 to xn.

Explore machine learning as function fitting with a housing example, predicting price from features like size, location, and age, and learn linear regression using Python notation.

Learn how to build a simple house price predictor in Python using linear regression in Google Colab, including data preparation, model fitting, and interpreting intercept and coefficients.

Build a house price predictor from size, location, and age, visualize the learned function, and analyze input effects while exploring the math of machine learning—algebra, linear algebra, calculus, and statistics.

Explore the basics of Python on Google Colab, review data types and variables, and introduce symbolic computation with SymPy, providing a gentle refresher for all levels.

Learn python basics in Google Colab, focusing on variables and data types—integers, floats, strings, and booleans—and printing with a separator for comma-separated output.

Learn Python arithmetic basics: addition, squaring, and division, and distinguish integers from floats while exploring integer division // and modulus for remainders.

Explore Python lists: ordered, mutable sequences with zero-based indexing, negative indexing, length queries, and appending elements. Learn how lists reference memory, creating side effects when sharing objects, unlike independent integers.

Learn how tuples are immutable, ordered data constructed with parentheses, and how to pack, unpack, and name values like name, age, and job for clear code.

Explore dictionaries as key-value data structures indexed by strings, using curly braces with names and scores; learn to handle missing keys with a default value and remove items with pop.

Explore how conditional statements power machine learning workflows using if, elif, and else with variables like y and threshold, including f-strings for dynamic output in Google Colab.

Explore Python data types - integers, floats, strings, booleans, lists, tuples, and dictionaries - and learn mutable versus immutable behavior and key-based indexing.

Convert data into numbers and treat objects as vectors in space, then measure distances and errors to minimize a cost function toward a local or global minimum.

Differentiate scalars from vectors, showing real numbers in R and vectors as ordered lists that scale by a scalar and combine through addition to describe features in n dimensions.

Represent students as feature vectors and measure similarity with Euclidean distance, using study hours, sleep, and exam scores; this connects coordinate differences to error and higher dimensional calculations.

Learn how k-nearest neighbors uses distance to classify data with two features: hours studied and sleep hours. The lecture demonstrates fitting a k-neighbors classifier and predicting outcomes for new students.

Represent data as vectors with features defining dimensions, and measure similarity by distance in high-dimensional geometry. Use k-nearest neighbors to learn from nearby points, linking scaling to fair distances.

Explore vectors as data points, their stacking into matrices for transformations. See how rows represent data points and columns represent features, and learn matrix operations like multiplication and transpose.

Explore transpose, matrix addition, and scalar multiplication, showing how to convert rows to columns, add matrices of the same shape, and scale entries for normalization and machine learning.

Learn how matrix multiplication performs a linear transformation, including rotation, scaling, and projection, mapping MxN to MxK, with 2x3 and 3x5 examples, and how chaining matrices appears in deep learning.

Define matrix multiplication and show A is m by n, B is n by p, yielding an m by p product. Relate to function composition with 2x2 and 2x1 examples.

Explore how 2x2 matrices act as linear transformations on point pairs, illustrating shearing, reflections, rotation, and scaling, and distinguish invertible from not invertible cases via the determinant.

We demonstrate matrix multiplication with 2 by 3 and 3 by 2 matrices, yielding a 2 by 2 result by multiplying rows and columns, verified with a computer tool.

Derive the formal definition of matrix multiplication by defining A as m-by-n, B as n-by-p, and C as AB, with Cij as the sigma sum of Aik Bkj from k=1 to n.

clarifies the formal definition of matrix multiplication, showing how a (m×n) and b (n×p) yield c (m×p) via c_ij = sum_k a_ik b_kj, and matches ab notation with an example.

Understand the identity matrix, which preserves a matrix, and how the inverse undoes transformations. Learn how the determinant signals invertibility and enables solving linear systems and the linear regression formula.

Explore the span of vectors through linear combinations of scaling and addition. See how alpha1 v1 plus alpha2 v2 equals any vector in the span, including the 2d plane.

Explore the norm of a vector as its length, computed by summing squares and taking square root, and see how it relates to distance, regularization, and normalization in machine learning.

Explore the determinant of a matrix as area in two dimensions, volume in three, and a test for invertibility; compute 2x2 det as ad-bc, and use Google Colab.

Explore the determinant of a square matrix, learn when a matrix is invertible versus singular, and see the 2×2 and 3×3 cases, including cofactor expansion and the first-row sign pattern.

Explore computing a 3x3 determinant via cofactor expansion along any row or column, note the checkerboard signs, and simplify with elementary row operations and determinant properties for machine learning.

Explore matrix algebra in Google Colab using NumPy to create, transpose, add, and multiply matrices. Demonstrates identity and inverse, determinant, singular matrices, rotation with radians and pi, and scalar multiplication.

Understand how linear models predict outcomes using weighted sums, where inputs are multiplied by weights and added with a bias, linking to linear algebra and linear regression.

Explore weighted sums in one and two variables using w x plus b, bias as a shift, and features projected onto weight directions, with practical NumPy demos.

Explore vector and dot product formulation for a weighted sum with X, W, and bias B, and see how Y = W^T X + B arises in linear models.

Relate data sets to matrix form by using an m-by-n data matrix X and predicting y_hat = Xw + b, illustrated with a two-feature, three-sample example.

Absorb bias into augmented vectors to simplify matrix multiplication, using X augmented with a ones column and weights W augmented with the bias, enabling W tilde transpose times X transpose.

Apply a linear model with y hat = x w + b, then pass the result through the sigmoid to yield probabilities between zero and one for logistic regression.

Understand functions and non-linearity in machine learning, distinguishing linear, affine, and non-linear functions and identifying common activation functions. See how models map inputs to outputs using non-linearities to boost learning.

Show how a function maps n-dimensional feature vectors to real-valued predictions, using weights and features, within the machine learning context.

Understand linear functions defined by matrices and how affine functions add a translation term, illustrating with simple examples like f(x)=3x and augmented variables that turn affine models into linear forms.

Explore nonlinear functions and why they violate additivity and homogeneity, with examples like x^2 and sin x, illustrating nonlinearity.

Show how linear models cannot capture saturation and diminishing returns, using studying versus exam performance to illustrate how curved relationships better describe real-world progress.

Learn how machine learning relies on numerics to optimize models by adjusting weights and biases to minimize loss using optimization algorithms, improving prediction accuracy.

Define learning with non-negative loss measuring prediction error between y and y hat, minimize the average loss over data to optimize theta using squared loss.

Learn how loss functions quantify prediction error, with squared loss as the sum of squared differences and the cost function J as the average loss over data to minimize.

Demonstrate real-world regression with squared loss using true vs predicted house prices, showing how squared differences amplify larger errors across data points.

Define a dataset-level cost function j(w,b) as the average squared errors between true values y_i and predictions w x_i + b over m examples, then minimize j(w,b).

Explore gradient descent as a formal method to minimize j by updating w and b along the negative gradient, using learning rate eta to control step size.

visualize optimization as geometry by treating the loss as a landscape where valleys mark good parameter values for weights and bias, and gradient descent follows the negative gradient to minima.

Demonstrates gradient descent on a one-feature regression model y = w x + b, with w and b initialized at 0, using eta over 200 epochs to update parameters.

Explore probability and uncertainty in machine learning, defining probability, illustrating conditional probabilities and Bayes' theorem, and recognizing that models implement Bayes' rule, with hands-on demonstrations in a Google CoLab notebook.

Explore the intuitive definition of probability and how it measures confidence in model predictions, using dice examples, threshold, and risk-aware decisions in machine learning.

Explore the formal definition of probability as a function p on a sample space s, with an event a in s, including conditional probability, intersections, and Bayes' theorem with normalization.

Explore conditional probability, the probability of an event given another event, and why it matters for predicting labels in machine learning with features and prior information.

Identify prior probabilities and likelihoods in a medical diagnosis, apply conditional probability and Bayes’ theorem to compute the probability of a positive result.

Explain how a proportionality constant relates two quantities, and how normalization yields posterior probabilities in Bayes' theorem by dividing by the normalization factor.

Link probability and classification in machine learning by outlining priors, likelihood, sensitivity, and posterior probability; apply Bayes' rule to predict the class from features.

Explore the normal (gaussian) distribution, its symmetry around the mean, and how standard deviation shapes spread. Learn about normalization to z-scores and practical probability insights like area under the curve.

Explore probability and uncertainty with code, computing conditional probabilities and applying Bayes' theorem to a medical test example. Visualize normal distributions, sample means and variances, and histogram plots.