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.

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

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.

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.

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.

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.

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 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.

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.

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.

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

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.

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.

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.

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.

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.

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.