# 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 (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?