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## Tentative Schedule

Week-by-week schedule will be posted here as the course progresses.
Week of Contents Sections
8/24 Linear systems and their geometric interpretation. Matrices and vectors. The matrix form of a linear system. Gauss-Jordan elimination. 1.1-1.2
8/31 Matrix vocabulary. Operations on matrices. Space Rn. Rank of a matrix. Number of solutions of a linear system.
Linear transformations from Rm to Rn. Matrix of a linear transformation. Linear transformations in a plane and how to visualize them.
The notion of inverse matrix.
1.3-2.1
9/7 More on geometry of linear transformations and their matrices. Geometric meaning of dot product. Projections and reflections in the plane and the 3-space; scalings, rotations, and shears. 2.1-2.2
Appendix A (no cross product)
9/14 Matrix multiplication and its meaning.
Inverse transformations and their matrices.
2.3-2.4
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.
3.1-3.2
9/28 Dimension of a subspace.
Geometric interpretation of rank (as dimension of image).
Rank-nullity 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.
3.3-3.4
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 1-3.
3.1-3.4, 4.1
10/12 Chapter 4: more general linear spaces.
4.1 Linear spaces and subspaces, span, basis
4.2 Linear transformations, isomorphisms, coordinates.
3.1-3.4, 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.
Gram-Schmidt process. Orthogonal matrices.
5.1, 5.2, 5.3