**Overview**

Linear algebra is the mathematics of phenomena that depend linearly on several parameters. Such phenomena can be modeled by **systems of linear equations**. We will discuss algorithms for computing solutions to systems of linear equations using **matrices**, rectangular arrays of numbers. Matrices can also be viewed geometrically as **linear transformations** on *n*-dimensional space. This viewpoint provides a segue to the notion of a linear transformation of an **abstract vector space**. We use **eigenvectors and eigenvalues** to describe the action of matrices and linear transformations in terms of simpler scaling actions. Finally we discuss a generalization of the dot product, the **inner product**, which encodes all the geometric information (length, distance, orthogonality) about a vector space, and, in particular, provides a notion of approximation in vector spaces.

**Course Documents**

**Course Notes**

- First Day Slides (Intro, 1.1, 1.2)
- Linear Systems (Section 1.1)
- Row Reduction (Section 1.2)
- Vector Equations (Section 1.3)
- Matrix Equations (Section 1.4)
- Solution Sets of Linear Systems (Section 1.5)
- Linear Independence (Section 1.7)
- Intro to Linear Transformations (Section 1.8)
- Linear Transformations of the Plane
- More on Linear Transformations (Section 1.9)
- Matrix Operations (Section 2.1)
- Matrix Inverses (Sections 2.2, 2.3)
- Matrix Factorizations (Section 2.5)
- Determinants (Section 3.1)
- Properties of Determinants (Section 3.2)
- Abstract Vector Spaces and Subspaces (Section 4.1)
- Null Space, Column Space, and Abstract Linear Transformations (Section 4.2)
- Linear Independence, Bases (Section 4.3)
- Coordinate Systems (Section 4.4)
- Dimension of a Vector Space (Section 4.5)
- Rank (Section 4.6)
- Intro to Eigenvectors and Eigenvalues (Section 5.1)
- Finding Eigenvalues and Diagonalizability (Sections 5.2, 5.3)
- Intro to the Inner Product (Section 6.1)
- Orthogonal Sets and Matrices (Section 6.2)
- Orthogonal Projections (Section 6.3)
- Least Squares Approximations (Section 6.5)
- Diagonalization of Symmetric Matrices (Section 7.1)
- Quadratic Forms (Section 7.2), Accompanying Mathematica Notebook

Goshen College Math