Key points of linear algebra
basics:
- basic concepts
- concrete interpretations
core:
- classifications
- structures (Lie theory)
- categorical
etc:
- algorithms
- applications
Basic concepts:
- elements: vectors
- objects: vector spaces
- maps: linear maps
Concrete egs:
- K^n
- polynomials and calculus
- analysis (continuous functions etc.)
vector/matrix interpretations
function/integral operators (with integration kernels)
[and reading matrix as columns, as rows, as total, as entries]
Subspaces, quotient spaces
affine spaces, cones, convex spaces
Classifications:
- vector spaces (dimension)
span, independent, basis
[and interpretations in terms of map / matrix]
- maps between spaces (rank; & rank-nullity etc.)
...and image, kernel, cokernel (im = ker coker = coker ker)
(nullity + rank = domain; rank + (conullity?) = target)
- structure of an operator
(+ Det, trace: --> char poly, Alt^k)
+ structure theorem for fg mods over PID
(and semisimplicity, simple / indecomposable)
+ structure of an operator (inv factor/primary decomposition)
+ ...and interpretation: eigen values, evectors / espaces / gen espaces
& Jordan form
- structure of an operator on hermitian space
+ spectral theorems
Lie theory: structures on a space:
(talk about Klein's Ehr... geometrization / symmetry program,
and origin in study of )
+ R/C/H/O (multiplications!) [and clifford algebras too]
- volume form: sl
- orthogonal form: so
- symplectic form: sp
[exotics]
real structures:
- orient, (spin) (also complex orientation)
- complex structure, quaternionic structure
...and (left) regular representations of complexes and quaternions
(-> representation theory)
- so(p,q): p+q space-time, physics
[and bases for the Lie algebras: rotations, boosts]
(and 6 degrees of freedom in affine 3-space, and pitch/roll/yaw on planes)
(cross product and physics love of quaternions & ijk basis,
b/c of Clifford algebras and 3+1 space-time)
[mention Jordan algebras too]
Categorical:
- representability of id (Hom(K,-) = id)
- dual (& covector interpretation!)
- Hom
- Tensor
(hard, weird; defined by universal property; multi-linear)
[also do concretely as "list of lists" = lex order basis]
- adjunction! $Hom(V,W) = V^* \otimes W$
- Schur functors
tensor powers, sym, alt, and others
--> representation thy of $S_n$, Schur-Weyl
...and tensor algebra, symmetric (polynomial) algebra, alt (grassmann) algebra
--> differential forms
(vector interpretation in R^3: grad/curl/div;
of course grad and div are general; curl is the weird one)
Algorithms:
- Gauss-Jordan
[- simplex method]
Applications:
- solving systems of equations
- calculus: derivative (as linear transform of tangent spaces) and chain rule (compose / mult matrices!) and gradient (it's dual: df is what's natural)
- PCA
- ???