Linear Algebra for Data Science and Machine Learning

Explore basic definitions of linear algebra, a subfield of algebra, focusing on variables, coefficients, and linear equations. Identify how coefficients can be real numbers and how linear equations form lines.

Learn to express a linear equation as a function of x by isolating y, analyzing slope, intercept, and coefficients for multiple variables with practical examples.

Explore how scalars, vectors, matrices, and tensors represent data in linear algebra. Learn their forms, dimensions, and implementations in Python with NumPy, TensorFlow, and PyTorch.

Install essential libraries in Google Colab, including sentence transformers, repl, matplotlib, numpy, pandas, TensorFlow, and PyTorch, import them, and verify their versions.

Explore implementing scalars and vectors using NumPy, showing that scalars are rank-zero tensors and vectors are rank-one arrays. Learn to create vectors with NumPy arrays and perform element-wise operations.

Develop and manipulate matrices, vectors, and tensors using numpy arrays, access elements with zero-based indexing, and perform multiplication; explore tensors in TensorFlow and PyTorch with shape, type, and numpy conversions.

Explore solving linear systems through a restaurant bill example, translating quantities and prices into matrix form, and analyzing intersections, overlaps, or parallel lines to identify unique, infinite, or no solutions.

Implement a linear system by plotting and solving for x2 as a function of x1, compare two and three couples, and visualize intersections to identify solutions or no solution cases.

Learn to represent data for machine learning by converting pandas data frames to numpy arrays, separating predictors x from the target y, and treating time series as 3d arrays.

Explore data representation with images as pixel grids and color channels, and learn how embeddings convert data to vectors that enable linear algebra for training and principal component analysis.

Represent data with pandas data frames, extract x and y variables, convert to numpy arrays, and explore a time-series air quality data set using expanddims to add a location dimension.

Create images and embeddings from a sentence transformer, visualize rgb image data as a pixel matrix, and build a simple tf.keras neural network with weights and a bias vector.

Explore vector representations and norms, including L1 (Manhattan) and L2 (Euclidean), and how lasso and ridge regularization affect linear regression.

Explore unit vectors and base vectors on Cartesian axes, express vectors as sums of base vectors, and examine projections, transposition, and orthogonal and orthonormal properties.

Learn to plot vectors using numpy, matplotlib, and regression algorithms in sklearn, visualizing with scatter plots and arrows on a cartesian plane, with axis limits and origin.

Calculate norms using NumPy's linalg.norm, defaulting to the l2 norm, with option to compute the l1 norm by setting r=1; l1 corresponds to Manhattan distance, l2 to Euclidean distance.

Demonstrate regularization by comparing linear regression with L1 (lasso) and L2 (ridge) norms on a three-feature dataset, showing coefficient sparsity and shrinkage.

Learn to distinguish unit and base vectors by calculating the L2 norm, normalizing a vector, and expressing it as a sum of base vectors i and j.

Learn how to transpose a vector using numpy functions such as np.dot, transpose, and vector.T, and how converting to a 2d row or column reveals the transpose.

Visualize the base vectors, compute dot products and L2 norms, and determine that i and j are orthogonal and orthonormal, while vector b may be orthogonal but not orthonormal.

Explore core matrix properties, including transpose, Frobenius norm, and the relation to vectors. Recognize symmetric, diagonal, and identity matrices and their computational efficiency in machine learning and NumPy.

Explore inverse matrices, determinants, cofactors, and the adjoint to compute inverses and understand singularity. Learn how orthogonal matrices simplify inverses via transpose.

Learn to implement matrices in Python with numpy and scipy, inspect shape and size, access elements by zero-based indices, and compute the Frobenius norm using numpy.linalg.norm.

Learn to implement transpose and symmetric matrices with numpy and scipy, compare A with its transpose, and verify symmetry using all.

Explore diagonal and identity matrices, extract diagonal elements with NumPy diag, verify diagonals, compute reciprocals, and obtain the inverse and identity matrices using NumPy functions.

Illustrate inverse computation with numpy linalg inv, verify the result with a dot product against the identity, and introduce the Moore-penrose pseudoinverse for singular or non-square cases.

Calculate the determinant by calling the linalg dot function with the scene matrix as parameter, yielding a scalar. Show that a singular matrix has determinant zero, via numpy.

Explore the properties of orthogonal matrices by showing rows and columns are orthonormal via l2 norm. Show that the transpose times the matrix equals the identity and the inverse.

Explore vector and matrix operations, including scalar broadcasting, addition, subtraction, and the Hadamard product, and learn reduction along axes with sums, means, minimum and maximum values.

Master matrix multiplication, including dimension compatibility and the distinction from elementwise Hadamard products, illustrated by a 3 by 3 result from row-by-column operations.

Learn how to perform scalar operations on arrays and a three by three matrix using NumPy in a Google Colab notebook, including addition, subtraction, multiplication, and division by a scalar.

Explore reduction operations using numpy, including sum, mean, prod, min, and max on arrays and matrices, with axis-based computations across rows or columns.

Apply element wise operations on tensors with matching dimensions to perform addition, subtraction, multiplication, and division across corresponding array positions, preserving the original shape.

Implement matrix multiplication using numpy matmul and verify shape compatibility. See how dimension mismatches trigger errors and how aligned shapes produce a successful result.

Demonstrate the dot product as element-wise multiplication and summation, verify it equals the L2 norm squared, and highlight its computational efficiency over the L2 norm.

Apply the cosine rule by computing the dot product and L2 norms, derive the angle with arccos, convert to degrees, and visualize the vectors in a 3d plot.

Explore how matrices transform vectors in Cartesian space through reflection, scaling, shearing, and rotation. See how determinants relate to area distortion and how inverses undo transformations in linear systems.

Explore eigenvectors and eigenvalues, showing how transformations scale or rotate vectors, with examples of unit orthonormal eigenvectors, symmetric matrices, and eigen decomposition, including determinant relationships and singular matrices.

Apply singular value decomposition to rectangular matrices, revealing left and right singular vectors and singular values, then use PCA for rotation and dimensionality reduction, including Moore-Penrose pseudoinverse for rectangular systems.

Explore linear transformations through reflection and scaling matrices, visualize vector mappings under A, B, and a scaling matrix, and learn how multiplication computes transformed coordinates.

Explore shearing and rotation matrices to transform vectors, rotate by ten degrees, and verify length preservation with the L2 norm.

Learn how determinants govern area changes under a transformation: apply a matrix to two vectors, compute original and transformed areas, and verify area scaling with determinant 0.53.

Create a transformation matrix and an array, multiply to transform, and use the inverse to recover the original values. Apply a consumption matrix to prices to compute the bill.

Explore eigenvectors and eigenvalues of a 2x2 matrix using NumPy to obtain eigenvalues, eigenvectors, and their transformation under matrix A, including scaling and orientation changes.

Explore eigenvectors and eigenvalues of singular and symmetric matrices, verify orthonormal eigenvectors, compute norms and dot products, and relate the determinant to the eigenvalues.

Demonstrate eigen decomposition of a matrix using NumPy to compute eigenvalues and eigenvectors, form the diagonal l and eigenvector matrix q, and reconstruct A via q l q inverse.

Learn how to apply singular value decomposition to rectangular matrices using numpy's svd, obtaining u, s, v h, reconstruct a via u, diag(s), v h, and verify with allclose.

Learn to compute the Moore-Penrose pseudoinverse for rectangular matrices using svd, form d_plus from reciprocal singular values, and apply u d_plus v h transposed to preserve shape.

Implement principal component analysis using numpy and sklearn to transform data, compute pc1 and pc2 variances, and interpret explained variance and mean-centered results.

Study linear systems using a restaurant example, representing data with matrices and coefficients; apply matrix multiplication, inverse, or Moore-Penrose pseudoinverse to solve for x.

Apply linear system implementation by constructing matrix a and vector y, computing the pseudo inverse a+, and solving for x to obtain item prices despite a being singular.

Explore how linear algebra underpins neural networks, using predictor and target variables to predict totals. Learn about weights, biases, batch processing, and tensor shapes.

Initialize neural network weights with a fixed seed, build w1, b1, w2, b2. Compute outputs via dot products and biases, compare to bill values, and minimize the error.

Apply eigendecomposition to decompose data into eigenvectors and eigenvalues and visualize eigenvectors as eigenfaces. Reconstruct any face via linear combinations of eigenfaces and use this approach for facial recognition.

Implement eigen decomposition on the Olivetti faces dataset to build and visualize eigenfaces, center data, compute covariance, and extract eigenvalues and eigenvectors.

Learn how singular value decomposition compresses data using the U, D, and V matrices to reconstruct images via matrix multiplication, with descending singular values guiding reconstruction and data savings.

Implement singular value decomposition to compress an image, convert to grayscale, reconstruct with varying singular values, and visualize how fewer values reduce data while preserving clarity.

Explore principal component analysis (PCA) for dimensionality reduction driven by variance. See how rotating data toward the first two components (PC1 and PC2) enables 2D visualization and simpler model training.

Demonstrates implementing PCA on the iris data set by standardizing features, transforming with PCA, and visualizing with pairplots, revealing that PC1 and PC2 capture most variance and separate species.

Explore how embeddings represent unstructured data as vectors and how cosine similarity measures image, text, and audio similarity; learn why orientation and angle indicate likeness, setting up the next lecture.

Encode 15 sentences into 384-number embeddings with the same model, then compute a cosine similarity matrix and visualize it as a heatmap showing sports, movies, and politics clustering.

Solve the homework exercises to recap linear systems, Google Colab data import, A and y in dataframe format, and GAN decomposition, SVD, PCA, and similarity between structured data.

Solve linear system exercises to predict house prices from features using A and y, then showcase a simple neural network with normalization to estimate nonlinear relationships.

Learn tf-idf text representation of technology and sports corpora, apply eigen decomposition and PCA for dimensionality reduction, compress images with SVD, and compare images using clip embeddings and cosine similarity.

Recap key linear algebra topics for data science and machine learning, including the four main data structures, linear systems, vectors, matrices, and transformations, followed by applied projects.

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