Linear Algebra by Jim Hefferon

Linear Algebra by Jim Hefferon - Linear Algebra Jim...

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Unformatted text preview: Linear Algebra Jim Hefferon ( 2 1 ) ( 1 3 ) fl fl fl fl 1 2 3 1 fl fl fl fl ( 2 1 ) x 1 · ( 1 3 ) fl fl fl fl x 1 · 1 2 x 1 · 3 1 fl fl fl fl ( 2 1 ) ( 6 8 ) fl fl fl fl 6 2 8 1 fl fl fl fl Notation R , R + , R n real numbers, reals greater than 0, n-tuples of reals N natural numbers: { , 1 , 2 , . . . } C complex numbers { . . . fl fl . . . } set of . . . such that . . . ( a .. b ), [ a .. b ] interval (open or closed) of reals between a and b . . . sequence; like a set but order matters V, W, U vector spaces v, w vectors 0, 0 V zero vector, zero vector of V B, D bases E n = e 1 , . . . , e n standard basis for R n β, δ basis vectors Rep B ( v ) matrix representing the vector P n set of n-th degree polynomials M n × m set of n × m matrices [ S ] span of the set S M ⊕ N direct sum of subspaces V ∼ = W isomorphic spaces h, g homomorphisms, linear maps H, G matrices t, s transformations; maps from a space to itself T, S square matrices Rep B,D ( h ) matrix representing the map h h i,j matrix entry from row i , column j | T | determinant of the matrix T R ( h ) , N ( h ) rangespace and nullspace of the map h R ∞ ( h ) , N ∞ ( h ) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha α iota ι rho ρ beta β kappa κ sigma σ gamma γ lambda λ tau τ delta δ mu μ upsilon υ epsilon nu ν phi φ zeta ζ xi ξ chi χ eta η omicron o psi ψ theta θ pi π omega ω Cover. This is Cramer’s Rule for the system x 1 + 2 x 2 = 6, 3 x 1 + x 2 = 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 x 1 times that, and equals the size of the final box. Hence, x 1 is the final determinant divided by the first determinant. Preface This book helps students to master the material of a standard undergraduate linear algebra course. The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The audience is also standard: sophmores or juniors, usually with a background of at least one semester of Calculus and perhaps with as much as three semesters. The help that it gives to students comes from taking a developmental ap-proach — this book’s presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and extensive, careful, exercises. The de-velopmental approach is what sets this book apart, so some expansion of the term is appropriate here. Courses in the beginning of most Mathematics programs reward students less for understanding the theory and more for correctly applying formulas and algorithms. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underly many mathematical investigations like elementary set and function facts, and a capac-ity for some independent reading and thinking. Linear algebra is an ideal spot ity for some independent reading and thinking....
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