1. Basic operations on vectors in the plane
2. Linear independence of vectors, basis vectors
3. Connections between algebraic operations on vectors and geometric quantities
4. Solutions of 2-by-2 systems of linear equations
5. Linear transformations of the plane and  matrix operations
6. Determinants and area
7. Characteristic polynomials and eigenvectors for 2-by-2 matrices

1. We will systematically extend all of the above algebraic and geometric ideas to the setting of vectors in 3-space.
2. We will briefly consider the extension of the above ideas to dimensions greater than 3.