# MATH 307 Homework 1 - Vocabulary List ## Linear Algebra Terms **Matrix-Vector Product**: The operation Ax where A is a matrix and x is a vector, computed using dot products of rows of A with the vector x. **Linear Transformation**: A function T(x) = Ax that maps vectors from one space to another while preserving vector addition and scalar multiplication. **Coefficient Matrix**: The matrix A in a system of linear equations Ax = b that contains the coefficients of the variables. **Determinant**: A scalar value det(A) computed from a square matrix that provides information about the matrix's properties (invertibility, area/volume scaling). **Co-domain**: The set of all possible output values of a function or transformation. **Standard Basis Vectors**: The unit vectors î and ĵ in 2D (or î, ĵ, k̂ in 3D) that form the canonical basis for the space. **Unit Cartesian Grid**: A coordinate system with unit spacing along perpendicular axes. **Eigenvalue (λ)**: A scalar λ such that Av = λv for some non-zero vector v, representing how much the eigenvector is scaled under the transformation. **Eigenvector (v)**: A non-zero vector whose direction remains unchanged under a linear transformation, only scaled by the eigenvalue. **Linearly Independent**: A set of vectors where no vector can be written as a linear combination of the others. **Basis**: A set of linearly independent vectors that span the entire vector space. **Scaled Eigenbasis**: A coordinate system formed by eigenvectors, where transformations become simple scaling operations. ## Calculus Terms **Taylor Series**: An infinite series representation of a function as a sum of terms calculated from the function's derivatives at a single point. **Maclaurin Series**: A special case of Taylor series expanded about x₀ = 0. **Taylor Polynomial (pₙ)**: The first n+1 terms of a Taylor series, providing a polynomial approximation to the function. **Taylor Remainder (Rₙ)**: The error term in Taylor's theorem representing the difference between the function and its Taylor polynomial. **Lagrange Form**: A specific representation of the Taylor remainder involving an unknown point ξ between x and x₀. **Convergent Series**: A series whose partial sums approach a finite limit. **Nontrivial Interval**: An interval containing more than just a single point. **Approximation Error**: The absolute difference between the true function value and its approximation. **Gradient (∇f)**: A vector of partial derivatives that points in the direction of steepest increase of a multivariable function. **Hessian Matrix (H)**: A square matrix of second-order partial derivatives of a multivariable function. **Local Maximum**: A point where a function achieves its highest value in a small neighborhood. **Minimax (Saddle Point)**: A point that is a local maximum in one direction and a local minimum in another direction. **Quadratic Approximation**: A second-degree polynomial approximation using the first three terms of a Taylor series. ## Geometric Terms **y-intercept**: The point where a line crosses the y-axis (where x = 0). **Unit-area**: A region with area equal to 1, often used as a reference for measuring area changes under transformations. **Invariant Direction**: A direction that remains unchanged under a transformation (characteristic of eigenvectors). **Proportionality Constant**: A scalar that relates two proportional quantities. **Surface Plot**: A three-dimensional visualization of a two-variable function z = f(x,y). **Markers**: Points highlighted on a graph to indicate specific locations of interest.