User:IssaRice/Linear algebra/Outline of linear algebra: Difference between revisions
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==First half of linear algebra== | ==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 [http://www.math.ucla.edu/~davidjekel/Linear_Algebra/Linear_Algebra_Equivalences.pdf this table]. | 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 [http://www.math.ucla.edu/~davidjekel/Linear_Algebra/Linear_Algebra_Equivalences.pdf this table]. | ||
Topics include: | Topics include: | ||
* Elementary row operations, elementary matrices | * Elementary row operations, elementary matrices | ||
* Row equivalence | * Row equivalence (several equivalent formulations) | ||
* Echelon form, reduced row echelon form, pivots | * Echelon form, reduced row echelon form, pivots | ||
* Linear independence, span, basis | * 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 | * Column space | ||
* Row space | * Row space | ||
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* Fundamental theorem of linear maps: rank + nullity = dimension of domain | * Fundamental theorem of linear maps: rank + nullity = dimension of domain | ||
* Equivalent properties of injective, surjective, bijective | * 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== | ==Second half of linear algebra== | ||
The second half focuses on operators and does inner product stuff | 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: | Topics: | ||
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* Norm | * Norm | ||
* Eigen stuff | * Eigen stuff | ||
* Determinants? | * Determinants? Trace? | ||
* Spectral theorem | * Spectral theorem | ||
* Singular value decomposition | * Singular value decomposition | ||
* Diagonalization | |||
* Orthogonality | |||
* Orthonormal bases | |||
* Orthogonal projection | |||
==Questions== | |||
* Can the second half be done first? | |||
Latest revision as of 21:28, 2 January 2019
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?