1 1 1 3 Linear Spaces Christopher Crawford

§ 1. 1. 1 -3 Linear Spaces Christopher Crawford PHY 311 2014 -01 -15

Outline • Linear (vector) space Linear combination Projection Geometry • Multilinear extensions: Metric (dot product) Exterior (cross) product Triple product Operators (next class) • ORTHOGONAL PROJECTION 2

Vector • Defining operation: LINEAR COMBINATION Structure • Basis: – Independent – Closure • Components – Array of coefficients Notation – Vector – Array – Einstein summation • Physical examples ? 3

Projection • Important course theme: longitudinal/transverse separation 4

Metric (inner or dot product) • Distance and angle; vector contraction (reduces dimension) 5

Orthogonal Projection (I) • A vector divides the space into parallel and orthogonal complements 6

Generalized Metric • For a basis which is not necessarily orthonormal 7

Exterior Products (wedge or cross) • Geometrically opposite to the inner product • Geometric significance – Perpendicular projection – AREA 8

Orthogonal Projection (II) 9

Triple product • 3 -dimensional object: Volume (of parallelepiped) 10

Exterior Algebra • Natural description of n-dimensional volume (area, volume) • By extension, the natural language of differential elements • Historical development of geometric algebra: – – Hamilton: quaternions (i, j, k) ij=k Grassman: exterior product Clifford: combined inner/exterior algebra (Pauli, Dirac matrices) Gibbs, Heaviside: simplified vectors with dot, cross product 11