MTH 5810 - Mathematical Methods for Machine Learning

• Describe linear and affine subsets of three-dimensional space both algebraically and geometrically
• Interpret the value of a dot product geometrically
• Interpret functions of two variables via the corresponding surface, level curves, and contour plot
• Perform elementary operations with matrices including manipulating expressions and equations involving matrices
• Determine if a matrix is invertible and find its inverse in this case
• Solve systems of linear equations using matrix methods and technology
• Express the solutions of a consistent matrix equation in parametric vector form
• Determine if a subset of n-dimensional space is a subspace
• Find the matrix of a linear transformation and bases for its kernel and range
• Show that a real number is an eigenvalue of a matrix with real number entries by finding the corresponding eigenvectors
• Determine the eigenvalues of a matrix using technology
• Use the inner product to find the length of a vector and the distance between two vectors
• Determine if a set of vectors is an orthogonal basis for a subspace of R^n
• Find the projection of a vector onto a subspace of R^n
• Find a general linear model by solving the normal equation
• Find and interpret the singular value decomposition of a matrix using technology
• Use numerical methods to locate extrema of a single-variable function
• Perform operations with series representations of the single-variable elementary functions
• Find first and higher-order partial derivatives of a function
• Interpret partial derivatives as rates of change in applications
• Find the total differential of a function of more than one variable, use it to estimate change
• Estimate error propagation using the total differential
• Construct the matrix form of the multivariate chain rule and use it to find a derivative
• Find the gradient of a function and interpret its direction
• Find directional derivatives of a function and interpret the results
• Determine the maximum, minimum, and saddle points on a surface
• Interpret mathematics from machine learning literature

• Single-variable calculus

• Geometry in three dimensions: lines, planes, distance, surfaces, vectors, and the dot product
• Matrix algebra
• Matrix equations and invertibility
• Real vector spaces
• Linear transformations and their matrices
• Inner products, orthogonality, projection, and applications
• Eigenvectors and eigenvalues and interpretation of matrix factorizations
• Review of single-variable differentiation
• Taylor series of elementary functions
• Finding critical points analytically and numerically
• Partial derivatives
• Gradients, directional derivatives, and the Jacobian
• Extrema of functions of two variables