## 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 $y \in Col(A)$, $y = Ax$ for some $x \in F^n$

### 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