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 systems of equations
• 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
• Spectral theorem
• Singular value decomposition