User:IssaRice/Linear algebra/Outline of linear algebra
Two approaches to linear algebra
- Coordinate-based approach: looks at concrete matrices, more emphasis on computation, works a lot in the standard basis. If linear algebra was analysis, this would be called "calculus"
- Coordinate-free approach: Abstract vector spaces, more emphasis on linear maps. If linear algebra was analysis, this would be called "real analysis".
First half of linear algebra
The point of the first half is to consider general linear transformations and classify them into injective/surjective/bijective. See this table.
Topics include:
- Elementary row operations, elementary matrices
- Row equivalence
- Echelon form
- Linear independence, span, basis
- Column space
- Row space
- Rank, column rank, row rank
- Linear systems of equations
- Matrix multiplication
- Null space = solution set of homogeneous linear system
- Reduced row echelon form
- Finding a basis for range, null space, range of transpose, null space of transpose
- Fundamental theorem of linear maps: rank + nullity = dimension of domain
- Equivalent properties of injective, surjective, bijective
Second half of linear algebra
The second half focuses on operators and does inner product stuff
Topics:
- Inner product
- Norm
- Eigen stuff
- Determinants?
- Spectral theorem
- Singular value decomposition