Linear Algebra for AI - Generative AI - Agentic AI

Compute vector size (magnitude) and direction from components, and explore the dot product as a scalar with commutativity and distributivity.

Translate the cosine rule into vector form, deriving x·y = |x||y| cos θ from |x−y|^2, and show how the dot product encodes direction—positive, zero, or negative.

Learn to compute projections from dot products: scalar projection, vector projection, projection onto a line via unit vectors, magnitudes, and cos theta.

Learn to create vectors as one-dimensional numpy arrays in Python, perform element-wise operations, compute dot products with numpy dot, and apply scalar multiplication in a hands-on Jupyter notebook.

Compute the L2 (Euclidean) norm to measure vector length, using 2D and 3D examples and numpy. Compare L1, P norms, and max norm, with connections to ridge and lasso regression.

Learn how to change the basis of a vector using orthogonal basis vectors, projecting A onto new basis x1 and x2 to obtain coefficients lambda1=2 and lambda2=3, so A=2x1+3x2.

Explore linear combinations and the span of two vectors, then define basis vectors as linearly independent sets that span a space, noting nonunit and nonorthogonal bases may require matrices.

Create matrices in Python using numpy, building 2d arrays and random matrices, and perform arithmetic operations such as addition, subtraction, hadamard product, dot product, and matrix multiplication.

Explore essential matrix types in linear algebra for ai: square, symmetric, triangular, diagonal, and identity matrices, with practical examples.

Explore matrix operations by performing transpose, swapping rows and columns, and observe shape changes; compute determinant with numpy and assess rank as the count of linearly independent rows or columns.

Explore how a two-by-two matrix solves a simultaneous equation ax=b by showing how basis vectors transform to the right-hand side, foundational for linear algebra in AI.

Explore how matrices transform vectors, using unit vectors and linear transformations to visualize geometry, solve systems, and see how matrix columns represent transformed basis vectors.

Explore how 2d vectors transform under matrices through scaling, rotation, and shear, with identity and diagonal cases, and apply these transformations in data science and facial recognition.

Explore how multiple matrix transformations—rotation, shear, and others—compose to transform a vector, and how the order of multiplication (YX vs XY) changes the result.

Master Gaussian elimination to solve linear systems, converting matrices to echelon form and applying back substitution or Gauss-Jordan to recover x, y, z.

Represent the linear system as matrices and vectors, and solve for x using the inverse, showing x equals a inverse times b and the identity matrix.

Show how the determinant scales area under a linear transformation; for a 2x2 matrix, det = ad - bc, and det = 0 when rows are dependent, blocking inversion.

Learn the Einstein summation convention for matrix transformations and the dot product, and see how repeated indices simplify coding with loops and reveal matrix multiplication as dot product.

Transform vectors between non-orthogonal bases using a transformation matrix and its inverse, and map coordinates between our world and the iPhone frame, contrasting with orthonormal bases and dot product methods.

Transform a vector between bases by applying the rotation matrix and its inverse to obtain coordinates in the iPhone world, illustrated with a 90-degree rotation.

Explore orthogonal matrices and orthonormal bases, where X^T X = I and the transpose serves as the inverse, enabling reversible data transformations and dot-product projections in data science.

Generate an orthonormal basis from linearly independent vectors using the Gram-Schmidt process. Use projections and normalization to form E1, E2, and E3 for transformations, while testing independence with determinants.

Learn to implement the Gram-Schmidt process in Python, producing an orthonormal basis from a matrix, and determine the dimension of the spanned space, handling linear dependence.

Explore the vector field concept, where arrows convey velocity magnitude and direction for particles and their neighborhoods, preparing you to study eigen vectors and gradient descent.

Learn how eigenvectors and eigenvalues describe how vectors transform under linear transformations, staying in the same span while some vectors stretch or shrink.

Derive eigenvalues and eigenvectors using the equation A x = λ x and det(A - λ I) = 0, with 2×2 examples and a rotation case that has no eigenvectors.

Learn to change basis with eigenvectors to diagonalize a transformation matrix, then compute powers by raising diagonal elements, and apply the inverse to recover the result.

Explore how changing to an eigenbasis diagonalizes a transformation, enabling quick power computations and practical applications in data science, page ranking with PageRank, image compression, and movie recommendations.

perform eigen decomposition on a square matrix with numpy, extract eigenvalues and eigenvectors, and reconstruct the original matrix using Q, L, and the inverse of Q.

Explore the basics of eigenvectors and eigenvalues through intuition on vectors and matrices, rotation and scaling, and how these concepts underpin neural network transformations.

Explore eigenvectors and eigenvalues through matrix transformations, showing how certain vectors stay in direction while scaling by the eigenvalue, and how to compute them in NumPy.

Learn how PCA uses eigenvectors and eigenvalues to reduce dimensionality by standardizing data, forming a covariance matrix, and projecting onto a lower-dimensional, orthogonal space that preserves maximum variance.

Learn linear discriminant analysis (LDA), a supervised dimensionality reduction method that preserves class separation by using both X and y, via scikit-learn.

Explore the Google PageRank algorithm by modeling a mini internet of pages and links, using a link matrix to iteratively compute page ranks as probabilities.

Implement the PageRank algorithm in Python using the power method to compute the eigenvalues and eigenvectors of the link matrix and converge on page rankings for a micro internet.

Apply damping in PageRank to handle pages with no outgoing links by mixing the link matrix with jumps, using M = D L + (1-D)/n J, demonstrated on page G.