Vector Spaces

Vector spaces are defined, along with the concept of subspace. Many examples are given, including Euclidean space and vector spaces of functions, including polynomials, differentiable and continuous functions (the latter topics are optional for students without a calculus background). Abstract linear transformations are defined, along with concepts such as the kernel and range. Linear combinations and linear independence are defined, and standard related theorems are proved. The concepts of span and basis are defined, along with the notion of coordinates of a vector with respect to a basis. Students may practice in the associated software the process of testing for linear independence and determining whether a given vector is in the span of some set of vectors. Basic examples of “standard” bases, such as those for Euclidean space, and spaces of matrices and polynomials, are considered. The Wronskian is treated briefly. The chapter concludes with an expanded list of properties that are equivalent to a matrix being invertible.