http://joshua.smcvt.edu/linearalgebra

Linear

Algebra

Notation R, R+ , Rn N, C (a .. b), [a .. b] ... V, W, U v, w, 0, 0V B, D, β, δ En = e1 , . . . , en RepB (v) Pn Mn×m [S] M⊕N ∼ V=W h, g H, G t, s T, S RepB,D (h) hi,j Zn×m , Z, In×n , I |T | R(h), N (h) R∞ (h), N∞ (h) real numbers, reals greater than 0, n-tuples of reals natural numbers: {0, 1, 2, . . . }, complex numbers interval (open, closed) of reals between a and b sequence; like a set but order matters vector spaces vectors, zero vector, zero vector of V bases, basis vectors standard basis for Rn matrix representing the vector set of degree n polynomials set of n×m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms, linear maps matrices transformations; maps from a space to itself square matrices matrix representing the map h matrix entry from row i, column j zero matrix, identity matrix determinant of the matrix T range space and null space of the map h generalized range space and null space

Lower case Greek alphabet, with pronounciation character α β γ δ ζ η θ ι κ λ µ name alpha AL-fuh beta BAY-tuh gamma GAM-muh delta DEL-tuh epsilon EP-suh-lon zeta ZAY-tuh eta AY-tuh theta THAY-tuh iota eye-OH-tuh kappa KAP-uh lambda LAM-duh mu MEW character ν ξ o π ρ σ τ υ φ χ ψ ω name nu NEW xi KSIGH omicron OM-uh-CRON pi PIE rho ROW sigma SIG-muh tau TOW as in cow upsilon OOP-suh-LON phi FEE, or FI as in hi chi KI as in hi psi SIGH, or PSIGH omega oh-MAY-guh

Preface

This book helps students to master the material of a standard US undergraduate ﬁrst course in Linear Algebra. The material is standard in that the subjects covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. Another standard is book’s audience: sophomores or juniors, usually with a background of at least one semester of calculus. The help that it gives to students comes from taking a developmental approach — this book’s presentation emphasizes motivation and naturalness, using many examples as well as extensive and careful exercises. The developmental approach is what most recommends this book so I will elaborate. Courses at the beginning of a mathematics program focus less on theory and more on calculating. Later courses ask for mathematical maturity: the ability to follow diﬀerent types of arguments, a familiarity with the themes that underlie many mathematical investigations such as elementary set and function facts, and a capacity for some independent reading and thinking. Some programs have a separate course devoted to developing maturity and some do not. In either case, a Linear Algebra course is an ideal spot to work on this transition. It comes early in a program so that progress made here pays oﬀ later but also comes late enough that students are serious about mathematics. The material is accessible, coherent, and elegant. There are a variety of argument styles, including direct proofs, proofs by contradiction, and proofs by induction. And, examples are plentiful. Helping readers start the transition to being serious students of mathematics requires taking the mathematics seriously so all of the results here are proved. On the other hand, we cannot assume that students have already arrived and so in contrast with more advanced texts this book is ﬁlled with examples, often quite detailed. Some books that assume a not-yet-sophisticated reader begin with extensive computations of linear systems, matrix multiplications, and determinants. Then, when vector spaces and linear maps ﬁnally appear and deﬁnitions and proofs start, the abrupt change can bring students to an abrupt stop. While this book

begins with linear reduction, from the start we do more than compute. The ﬁrst chapter includes proofs showing that linear reduction gives a correct and complete solution set. Then, with the linear systems work as motivation so that the study of linear combinations is natural, the second chapter...