Algebra lineal

Linear Algebra
Jim Hefferon

1 3

2 1

1 3

2 1
x1 · 1 3

2 1

x1 · 1 2 x1 · 3 1
6 8

2 1

6 2 8 1

Notation R N C {. . . . . .} ... V, W, U v, w 0, 0V B, D En = e1 , . . . , en β, δ RepB (v) Pn Mn×m [S] M ⊕N V ∼W = h, g H, G t, s T, S RepB,D (h) hi,j |T | R(h), N (h) R∞ (h), N∞ (h) real numbers natural numbers: {0, 1, 2, . . .} complex numbers set of . . . such that . . .sequence; like a set but order matters vector spaces vectors zero vector, zero vector of V bases standard basis for Rn basis vectors matrix representing the vector set of n-th degree 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 hmatrix entry from row i, column j determinant of the matrix T rangespace and nullspace of the map h generalized rangespace and nullspace

Lower case Greek alphabet name alpha beta gamma delta epsilon zeta eta theta character α β γ δ ζ η θ name iota kappa lambda mu nu xi omicron pi character ι κ λ µ ν ξ o π name rho sigma tau upsilon phi chi psi omega character ρ σ τ υ φ χ ψ ω

Cover. This isCramer’s Rule for the system x1 + 2x2 = 6, 3x1 + x2 = 8. The size of the first box is the determinant shown (the absolute value of the size is the area). The size of the second box is x1 times that, and equals the size of the final box. Hence, x1 is the final determinant divided by the first determinant.

Preface
In most mathematics programs linear algebra comes in the first or second year,following or along with at least one course in calculus. While the location of this course is stable, lately the content has been under discussion. Some instructors have experimented with varying the traditional topics and others have tried courses focused on applications or on computers. Despite this healthy debate, most instructors are still convinced, I think, that the right core material is vectorspaces, linear maps, determinants, and eigenvalues and eigenvectors. Applications and code have a part to play, but the themes of the course should remain unchanged. Not that all is fine with the traditional course. Many of us believe that the standard text type could do with a change. Introductory texts have traditionally started with extensive computations of linear reduction, matrixmultiplication, and determinants, which take up half of the course. Then, when vector spaces and linear maps finally appear and definitions and proofs start, the nature of the course takes a sudden turn. The computation drill was there in the past because, as future practitioners, students needed to be fast and accurate. But that has changed. Being a whiz at 5 × 5 determinants just isn’t important anymore.Instead, the availability of computers gives us an opportunity to move toward a focus on concepts. This is an opportunity that we should seize. The courses at the start of most mathematics programs work at having students apply formulas and algorithms. Later courses ask for mathematical maturity: reasoning skills that are developed enough to follow different types of arguments, a familiarity with thethemes that underly many mathematical investigations like elementary set and function facts, and an ability to do some independent reading and thinking. Where do we work on the transition? Linear algebra is an ideal spot. It comes early in a program so that progress made here pays off later. But, it is also placed far enough into a program that the students are serious about mathematics, often majorsand minors. The material is straightforward, elegant, and accessible. There are a variety of argument styles—proofs by contradiction, if and only if statements, and proofs by induction, for instance—and examples are plentiful. The goal of this text is, along with the presentation of undergraduate linear algebra, to help an instructor raise the students’ level of mathematical sophistication. Most...