Stanford UniversityMATH 51math 51 notes 2007

Math 51 TA notes — Autumn 2007 Jonathan Lee December 3, 2007 Minor revisions aside, these notes are now essentially final. Nevertheless, I do welcome comments! Go to http://math.stanford.edu/~jlee/math51/ to find th ...

Math 51 TA notes — Autumn 2007 Jonathan Lee December 3, 2007 Minor revisions aside, these notes are now essentially final. Nevertheless, I do welcome comments! Go to http://math.stanford.edu/~jlee/math51/ to find these notes online. Contents 1 Linear Algebra — Levandosky’s book 2 1.1 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Linear Combinations and Spans . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Dot Products and Cross Products . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.7 Matrix-Vector Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.8 Nullspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.9 Column space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.10 Subspaces of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.11 Basis for a Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.12 Dimension of a Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.13 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.14 Examples of Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 7 1.15 Composition and Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . 8 1.16 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.17 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.21 Systems of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.23 Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.25 Symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.26 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 12 Vector Calculus — Colley’s book 11 2.2 Differentiation in Several Variables . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 The derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.4 Properties of the Derivative; Higher Order Derivatives . . . . . . . . . 15 2.2.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.6 Directional Derivatives and the Gradient . . . . . . . . . . . . . . . . 15 2.3 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Parametrized curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Maxima and Minima in Several Variables . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Differentials and Taylor’s Theorem . . . . . . . . . . . . . . . . . . . 16 2.4.2 Extrema of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.3 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.4 Some Applications of Extrema . . . . . . . . . . . . . . . . . . . . . . 19 1 Linear Algebra — Levandosky’s book • useful (non-)Greek letters: α, β, γ, δ, u, v, x, y, z 1.1 Vectors in Rn • a vector in Rn is an ordered list of n real numbers; there are two basic vector operations in Rn: addition and scalar multiplication • examples of vector space axioms — commutativity, associativity — “can add in any order” • standard position — vector’s tail is at the origin 1.2 Linear Combinations and Spans • a linear combination of vectors {v1, . . . , vk} in Rn is a sum of scalar multiples of the vi • the span is the set of all linear combinations • a line L in Rn has a parametric representation L = {x0 + αv : α ∈ R} with parameter α • given two distinct points on a line, we can find its parametric representation; can also parametrize line segments 2• two non-zero vectors in Rn will span either a line or a plane; the former happens if they’re collinear (or one is redundant) • a plane P in Rn has a parametric representation P = {x0 + αv1 + βv2 : α, β ∈ R} with parameters α and β • given three non-collinear points on a plane, we can find its parametric representation • checking for redundancy — “row reduction” 1.3 Linear Independence