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

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

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

4. Projection
Important course theme: longitudinal/transverse separation

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

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

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

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

9. Orthogonal Projection (II)

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

11. 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