Abstract Vector Spaces
The Definition of a Vector Space
Subspaces
Bases and Dimension
Linear Independence, Spanning Sets, and Bases
Dimension
Obtaining Bases
Linear Transformations
Linear Transformations Between Abstract Vectors
Rank and Nullity
Linear Maps as Matrices
Column space / Nullspace
If , for some
Change of Coordinates
Isomorphisms of Vector Spaces
Diagonalizability
Eigenvectors and Diagonalization
Diagonalization
Applications of Diagonalization (Not in Final)
Inner Product Spaces
Inner Products
Orthogonality
Orthonormal Bases
Projections
The Gram-Schmidt Orthogonalization Procedure
Orthogonal Complements
Projection onto a Subspace
Application: Method of Least Squares (Not in Final)
Least Squares Curve Fitting
Orthogonal Diagonalization
Orthogonal and Unitary Matrices
Matrix is orthogonal if its columns are orthonormal
Or Matrix is orothogonal if QQ^T=I
Properties of the Adjoint
Schur’s Triangularization Theorem
Orthogonal and Unitary Diagonalization of Matrices
Orthogonal and Unitary are the same, but R for orthogonal and C for unitary
Similarly, Symmetric and Hermitian
