# User:IssaRice/Linear algebra/Outline of linear algebra

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## Contents

## 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 (i.e. does not restrict to operators) and classify them into injective/surjective/bijective. See this table.

Topics include:

- Elementary row operations, elementary matrices
- Row equivalence (several equivalent formulations)
- Echelon form, reduced row echelon form, pivots
- Linear independence, span, basis (there seems to be a choice of doing this as sets vs lists, although I think even books that use sets are sometimes forced to then define an "ordered basis")
- Column space
- Row space
- Rank, column rank, row rank
- Linear systems of equations
- Matrix multiplication
- Null space = solution set of homogeneous linear system
- 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
- Change of basis
- Matrix similarity
- Going back and forth between coordinate-based and coordinate-free approach, e.g. diagonalization via multiplying on both sides vs diagonalization via picking some basis

## Second half of linear algebra

The second half focuses on operators (linear maps that map from a vector space to the same vector space) and does inner product stuff. Maybe this is called "spectral theory".

Topics:

- Inner product
- Norm
- Eigen stuff
- Determinants? Trace?
- Spectral theorem
- Singular value decomposition
- Diagonalization
- Orthogonality
- Orthonormal bases
- Orthogonal projection

## Questions

- Can the second half be done first?