Week of 
Contents 
Sections 
8/24 
Linear systems and their geometric interpretation. Matrices and vectors. The matrix form of a linear system. GaussJordan elimination. 
1.11.2 
8/31 
Matrix vocabulary. Operations on matrices. Space R^{n}. Rank of a matrix.
Number of solutions of a linear system. Linear transformations from R^{m} to R^{n}. Matrix of a linear transformation.
Linear transformations in a plane and how to visualize them. The notion of inverse matrix. 
1.32.1 
9/7  More on geometry of linear transformations and their matrices. Geometric meaning of dot product.
Projections and reflections in the plane and the 3space; scalings, rotations, and shears.

2.12.2 Appendix A (no cross product) 
9/14  Matrix multiplication and its meaning.
Inverse transformations and their matrices.

2.32.4 
9/21  Kernel and Image of a linear transformation.
Subspaces, span of vectors, linear (in)dependence. Basis. Finding Image (as span of column vectors)
and kernel (solve the system!) of a linear transformation.

3.13.2 
9/28 
Dimension of a subspace.
Geometric interpretation of rank (as dimension of image).
Ranknullity theorem and how to use it.
Coordinates of a vector with respect to a basis.
Finding a matrix of linear transformation in a new basis.

3.33.4 
10/5 
More on invertible matrices.
Further applications/discussion of Chapter 3.
Begin Chapter 4: more general linear spaces are useful!
Exam on 10/9 covering Chapters 13.

3.13.4, 4.1 
10/12 
Chapter 4: more general linear spaces.
4.1 Linear spaces and subspaces, span, basis
4.2 Linear transformations, isomorphisms, coordinates.

3.13.4, 4.1 
10/19 
Linear transformations in arbitrary vector spaces,
isomorphisms, the matrix of a transformation in a particular basis.

4.2, 4.3 
10/26 
Orthogonality, orthogonal projections, orthonormal basis.
GramSchmidt process. Orthogonal matrices.

5.1, 5.2, 5.3 