Math 0-1: Linear Algebra for Data Science & Machine Learning

Review lines and planes via y = mx + b, where x is input and y is output, and relate to ax + by + c = 0.

Review two-equation, two-unknown systems by equating right sides to solve for x and y, then analyze cases: intersection (one solution), parallel (no solution), and identical lines (infinite solutions).

Gaussian elimination provides a systematic method to solve linear systems via row operations: multiply a row by a nonzero constant, add or subtract rows, and swap rows.

Explore linear systems of three equations and three unknowns, using Gaussian elimination to detect contradictions and show no solution exists when the equations cannot be satisfied simultaneously.

Examine systems of three equations in three unknowns that have infinite solutions when two planes coincide, producing a line of intersection, analyzed via Gaussian elimination and coefficient table.

We review solving linear systems—from two and three equations to n equations—using Gaussian elimination with elementary row operations, and distinguish unique solutions from no solution or infinite solutions.

Vectors have magnitude and direction and can be represented as coordinates or vectors; in data science and ML, data points are vectors used to separate classes with a decision boundary.

Add and subtract vectors by using element-wise sums of equal-size vectors, then convert between magnitude-angle and Cartesian components using cos, sin, and arctan.

Explore the dot product, the vector multiplication that yields a scalar, equal to |x||y|cosine(theta). See its applications in portfolios, neural networks, and the hyperplane concept.

Learn how dot product properties—commutative, distributive, associative, and the magnitude relation |v|^2 = v·v—work, and why x·y = |x||y|cos theta follows from the definition.

Practice dot product calculations in NumPy using Python, learn vector magnitudes and element-wise versus dot product, and compute angles between vectors with dot products and inverse cosine.

Explore how dot products generate embeddings in neural networks and apply cosine similarity to measure distances between vectors, comparing Euclidean and cosine distances for clustering and face recognition.

Compute the unit vector by dividing x by its magnitude to obtain x hat; ensure length equals one and direction matches x, as shown by the dot product proof.

Understand matrices as tables of numbers and linear transformations, and learn how matrix multiplication with vectors enables modeling datasets, neural network weights, and graph representations.

Master element-wise matrix addition and scalar multiplication on same-size matrices, preserving shape, and view subtraction as addition with a negative scalar.

Explore matrix multiplication as a generalization of the dot product, learn inner-dimension rules, note its non-commutativity, and work through scalar and matrix cases with examples.

Explore matrix vector products as a simplified matrix multiplication, where A is N by D and x is D by 1, yielding N by 1 result through a_i dot x.

Apply the matrix-vector product to compute neural network layer outputs, using a weight matrix and input vector to produce multiple features via the dot product.

Explore the element-wise product, a pointwise operation for matrices of the same size, symbolized by a circle with a dot, used in deep learning and machine learning.

Explore outer product versus inner (dot) product, showing how two vectors form a matrix or a scalar, and compare computational efficiency: dot product then scale is cheaper.

Explore a cutting-edge application: learning a projection matrix to map image and text vectors for vision-language models, enabling transfer learning with cosine similarity.

Explore matrix concepts in python with numpy, comparing arrays and matrices, performing matrix multiplication and dot products, and using transpose and reshape to create column and row vectors.

Represent systems of linear equations as the matrix-vector product AX = B, with A as the coefficient matrix, X as the variable vector, and B as constants.

Review vectors and matrices, including dot and inner products with magnitude and cosine. Cover element-wise operations, transpose, matrix multiplication, outer products, and ax = b.

Explore diagonal matrices, square matrices with zeros on the off-diagonals, including the identity, and how they act as stretching transformations with easy inverse and transpose in probability, statistics, and SVD.

Explain what an inverse is and why the inverse of the inverse returns the original matrix, and relate this to inverse functions like exp and log.

Explore singular matrices, whose inverses do not exist, learn why they resemble dividing by zero, and see how infinite or no solutions arise when solving Ax = b.

Learn the matrix transpose, its shape rules, and dimensional analysis, and see how it enables vector-matrix products like W^T X in neural networks and batch computations.

Explore the five matrix transpose properties, including distribution over addition, scalar factors, reverse order in products, and inverse transpose, with applications to multivariate Gaussian expressions in data science.

Explore how the transpose generalizes to higher-dimensional tensors by permuting dimensions, illustrated via code like transpose(210) to align images and kernels across deep learning libraries.

Explore orthogonal matrices by proving that u^T u = I implies u u^T = I. Practice manipulating matrices to confirm rows and columns are orthonormal.

Explore the inverse of a product and verify that (AB)^{-1} = B^{-1} A^{-1}. Multiply by AB to obtain the identity, showing a rule similar to the transpose.

Show that for a symmetric matrix A, the transpose of the inverse equals the inverse of the transpose, and since A is symmetric, the inverse is also symmetric.

Explore orthogonal (unitary) matrices and prove they preserve vector length and angles, using u^T u = I and the transformed inner product to show length and angle invariance.

Explore the determinant of a 2x2 matrix, compute ad minus bc, and identify singular, noninvertible cases; connect the determinant to the parallelogram area.

Compute determinants for any n by n matrix using minors and cofactors, following the plus-minus pattern, and apply a recursive approach suitable for computer implementation.

Explore determinant identities, including det(CA)=C^n det(A) for n×n matrices and det(A^T)=det(A). Learn how det(AB)=det(A)det(B), det(A^{-1})=1/det(A), and det(A^n)=(det A)^n.

Show that the determinant's magnitude is one for orthogonal or unitary matrices by using U U^T = I, U^T U = I, and det(U^T) = det(U).

Compute the matrix trace as the sum of a matrix's diagonal elements, a scalar used in Bayesian machine learning alongside the inverse Wishart distribution and the quadratic form.

Learn how positive and negative definite matrices shape quadratic forms x^T A x, identify semidefinite and indefinite cases, and relate these to covariance in probability.

Practice NumPy matrix operations by inverting matrices, handling diagonal and near-singular cases, and transposing tensors, then verify orthogonality, compute determinants, and explore definiteness via quadratic forms.

Navigate matrix operations and special matrices, including the identity and diagonal matrices, inverses via Gaussian elimination, solve ax=b, and cover transpose rules, determinants, and definiteness classifications.

Explore linear independence and dependence in matrix equations, understand when systems have unique, infinite, or no solutions, and see how rows, columns, and linear combinations affect invertibility and determinant.

Explore the geometric interpretation of linear combinations in three dimensions, showing how three planes can be parallel or intersect, yielding no or infinite solutions.

Explore matrix rank as the number of linearly independent rows and columns. Note that column rank equals row rank and full rank denotes maximal independence.

Explore four matrix decompositions—Cholesky, LU, QR, and SVD—and their factorizations. Learn how SVD enables low-rank approximations and its connection to PCA, with applications in data science.

Explore how the rank of AB relates to the ranks of A and B, showing the rank cannot exceed the smaller rank and illustrating low rank approximations.

Explore low rank approximations for large matrices by factorizing A into two skinny matrices, yielding a hat A with rank K, and assess the fit using the Frobenius norm.

Leverage SVD for low-rank approximations and dimensionality reduction, revealing the most important features for data visualization and rank estimation through best rank-k projections.

Explore low rank adaptation (LoRA) for diffusion models and large language models, applying lightweight low-rank updates to neural network weights to accelerate training and adaptation.

Form A^T A and A A^T to obtain symmetric, positive semidefinite matrices, and explain that they may fail to be positive definite if A is not full rank.

Clarify the relationship between rank and positive definiteness, showing that if A is not full rank, A^T A and A A^T are not positive definite, with zero eigenvalues.

Explore checking matrix rank with numpy and Python, and learn how SVD reveals rank, orthogonality, and rank-one and rank-two approximations in practice.

Analyze linear independence and rank, define full rank and rank deficiency, and bound rank by min(m, n); apply SVD for the best low rank approximation with U, S, V^T.

Explore eigenvalues and eigenvectors: learn how Av = lambda v keeps direction, identify eigenvalues via det(A - lambda I) = 0, and compute eigenvectors for each lambda.

Learn why eigenvalues and eigenvectors satisfy A v = λ v by using the characteristic polynomial f(λ) = det(A − λI) and the condition det(A − λI) = 0.

Examine rotation matrices as orthogonal transforms, verify rotation by theta, derive eigenvalues cos theta ± i sin theta, and compute corresponding complex eigenvectors.

Show that A^T A and A A^T share the same eigenvalues by multiplying the eigenvalue equation by A^T and setting y = A^T x.

Explore the conjugate transpose (hermitian) of complex matrices, and connect it to inner products, quadratic forms, unitary matrices, and vector length, with roots in eigenvalues and vectors.

Explore why hermitian or real symmetric matrices have real eigenvalues, using the eigenvalue equation and the quadratic form v^H a v over v^H v, and why complex numbers are essential.

Show that hermitian and real symmetric matrices have orthogonal eigenvectors by comparing eigenpairs and using the inner product, enabling principal components analysis and matrix decompositions when eigenvalues differ.

Demonstrate diagonalization by assembling eigenvectors and eigenvalues into matrices V and Lambda, showing A V = V Lambda and A = V Lambda V^{-1}, with distinct eigenvalues guaranteeing independent eigenvectors.

Show that positive definite matrices have positive eigenvalues, and use diagonalization a = v Λ v^T with y = v^T x to test definiteness via sum λ_i y_i^2.

Explore how to compute the determinant from eigenvalues using diagonalization, showing that det(A) equals the product of its eigenvalues.

Explore how eigenvalues determine matrix invertibility, using the determinant and the product of eigenvalues; show positive definite matrices are invertible and apply to covariance matrices with shrinkage or bayesian estimates.

Construct the SVD by diagonalizing A^T A as V Λ V^T, define Σ from the eigenvalues, and show A = U Σ V^T with U = A V Σ^{-1}.

Learn how to compute matrix powers efficiently through diagonalization using eigenvalues and eigenvectors, and apply this to forecasting future states and Markov models.

Explore the vanishing gradient problem in neural networks, showing how repeated matrix multiplications during backpropagation shrink gradients and hinder training in deep architectures.

Explore functions of matrices through power series and diagonalization, computing matrix exponentials and other functions by applying scalar functions to eigenvalues, with applications in statistics and control theory.

Explore how the Fibonacci sequence, defined by a recurrence relation, emerges from a linear transformation and eigenvalues and eigenvectors to yield a closed-form, constant-time expression for y_n.

Explore eigenvalues and eigenvectors with NumPy, practice by-hand calculations, and demonstrate diagonalization, matrix powers, and fractional powers, while linking eigenvalues to rank and positive definiteness.

Explore why a matrix may lack a real square root when eigenvalues are negative. Cast to complex in numpy to compute the square root via diagonalization.

Explore how eigenvalues and eigenvectors solve AV = λV, use the characteristic polynomial to find them, and diagonalize A as A = V Λ for powers and functions.

Explore the appendix and FAQ in this course, showing their role as optional supplementary material with common questions, and invite students to ask on the Q&A.

Understand the pre-installation approach as scalable guidelines that emphasize learning principles over syntax. See how Python setup and pip use, with examples like Theano, CNTK, and OpenAI gym, illustrate prerequisites.

Learn how to install data science libraries on Windows with Anaconda, isolating environments and managing Python versions, then install TensorFlow, Keras, PyTorch, scikit-learn, CNTK, and OpenAI Gym.

Set up a cross-platform data science environment by installing numpy, scipy, matplotlib, pandas, ipython, Theano, and TensorFlow; run Ubuntu on a VirtualBox virtual machine on Windows or Mac.

Learn to access the course Python notebooks via the provided code link by logging in, since notebooks are not on GitHub, and follow VPN and repo navigation cautions.

Learn GitHub basics and practical coding practices by downloading the course repo with git clone or the download button, and complete hands-on exercises across Colab, Jupyter, and IDEs.

Explore the core math prerequisites for machine learning and data science—calculus, linear algebra, and probability—and learn the recommended order, including matrix calculus and practical applications.

Explore whether YouTube can teach calculus, and why problem solving, not entertainment, builds real math intuition for machine learning prerequisites.

Balance academic rigor with practical machine learning by clarifying prerequisites, defining beginner and expert perspectives, and demonstrating code-first algorithms on real-world data and 2D datasets.

Discover how to sequence machine learning topics through course dependencies, starting with linear regression, then logistic regression, and progressing toward neural networks and deep learning.

Navigate the course path by understanding prerequisites and the relationships among supervised, unsupervised, and reinforcement learning. Connect word embeddings, RNNs, CNNs, clustering, EM algorithm, and other concepts to guide progression.

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