Below is a list of topics and concepts that will be used throughout the course. They should be part of your undergraduate curriculum, but it is frequently the case that these concepts have not been used recently, especially by CS majors. So you may need to freshen up. This list provides a guideline to do so. 1. Linear algebra.- linear (vector) spaces: what they are, how they are represented, how they are transformed one onto the other- linear combinations: what it takes for a space to be a linear space.- linear independence: what it means, what happens when a set of vectors is not linearly independent; how an it be transformed into another set that is "equivalent" and yet is made of linearly independent elements- basis: what it is, how to transform from one to the other- inner products, orthogonality: what it means; how to transform from an inner product to another; abstract definition, generalization to vector spaces beyond R^n- orthogonal matrices: definitions, properties- Gram-Schmidt orthogonalization: procedure and its meaning- range space, null space, rank: definitions, interpretation, calculation: How do you compute the (range/null/rank) of a matrix?- eigenvalues, eigenvectors: definitions, interpretation; spectrum of a matrix- symmetric matrices: definitions, properties- solving linear systems of equations of the form Ax = b: you must be able to solve a linear system of equations by hand, provided it is solvable. You must also be able to ...- conditions under which the system above can be solved, and can be solved uniquely: determine whether a linear system of equation is solvable (has at least one solution), and whether it has multiple solutions; in the latter case, you must be able to represent the set of all possible solutions analytically- least-squares approximation to the solution of a linear system: If a linear system does not admit a solution, how do you "relax" it so that it admits infinitely many, and how can you determine a criterion to choose one particular solution, among the infinitely many, that is in some sense "sensible".- (optional: SVD)2. Calculus and numerical analysis.- limits: you must understand the concept of limit; epsilon-delta and all that; be facile with computing limits and derivatives- differentiation- Riemann integral: how to compute the integral of a function that is represented using samples, as opposed to analytically.- Gauss-Newton methods, conjugate gradient, Levemberg-Marquardt: how to minimize a scalar-valued smooth function iteratively; conditions under which you are guaranteed to converge to the global minimum; necessary conditions for a local minimum- Constrained optimization, Lagrangian multipliers: how to frame a constrained optimization problem into an unconstrained one.- trigonometric functions, Fourier series: series of functions (polynomials, sin/cos, exponentials)- solution of linear system of ordinary differential equations x-dot = A x; you must be able to solve a linear system of equations with constant coefficients; you must have a procedure to compute, or at least approximate, matrix powers and matrix exponentials3. Basic probability and stochastic processes- probability space: what it is, examples- random variables: definition, interpretation, understand the difference between a random variable, its probabilistic description (distribution, density, moments etc.), and its realization (sample)- expectation: definition, interpretation, examples- conditional expectation: definition (it is a random variable), interpretation- moments: definition, interpretation; second- and third-order moments- marginalization: how to "integrate out" a random variable.- Bayes' rule: definition, interpretation- law of large numbers, central limit theorem: definition, interpretation, implications- stochastic process: definition, interpretation, understand the difference between a random process, its probabilistic description, its realizations- correlation function: definition, interpretation