§1.1. 1-3 Linear Spaces Christopher Crawford PHY 311 2014-01-15.

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Presentation transcript:

§ Linear Spaces Christopher Crawford PHY

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 4 Important course theme: longitudinal/transverse separation

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