Here's what I plan to cover each day in the semester. 

• 1/26: 1.1 Solving systems of linear equations using row operations.
  
• 1/28: 1.2 (Reduced) row echelon form.
• 1/31: 1.3 Vectors in Rⁿ, linear combinations, the Span.  The Span of a set of vectors is a very important concept.
• 2/2: 1.4 Matrix multiplication as a linear combination of the columns. Matrix multiplication as dot products.  (The  matrix multiplication we talk about here is Ax where A is a matrix and x is a vector. Later in 2.1 we will see how to multiply two matrices.)  Solving Ax=b.  Among other things, we learned that for a given A, the equation Ax=b will always have a solution if A has a pivot in every row.  If A does not have a pivot in everey row, then for some b, Ax=b will have a solution but for most b, there will be no solution.
• 2/4: 1.5 Homogeneous systems, Ax=0.  We learned that if A has a pivot in every column, then Ax=0 has only the trivial solution x=0.  But if some columns of A have no pivot, there are nontrivial solutions to Ax=0.
• 2/7: 1.7 Linear independence- this is a very important concept.
• 2/9: 1.8  Linear transformations from Rⁿ to R^m. 
• 2/11: 1.9 The matrix of a linear transformation (this is a special case of something we will see later when we study bases).
• 2/14: 2.1 Operations on matrices- addition and multiplication, and transpose.
• 2/16: 2.2 Another matrix operation, the inverse.  No office hours today.
  
• 2/18: 2.3 Not all matrices have an inverse, here we look at a number of criteria for determining when a matrix is invertible.  Yes, there are lots of criteria, but invertibility is an important and useful concept so we dwell on it a bit.
• 2/21: 2.4 Partitioned matrices.
• 2/23: 2.5 the LU decomposition.
• 2/25: 3.1, 3.2, 3.3 Determinents.  We will just skim over this chapter.  We'll focus on the properties of determinents such as det(AB) = det(A) det(B) and det(A^T) = det(A) and not worry about why they are true or even how to compute the determinent by hand except in the 2x2 and 3x3 cases.
  
• 2/28: Review
• 3/2: Exam #1 on sections 1.1-1.5,1.7-1.9, 2.1-2.5
• 3/4: 4.1 Vector spaces and subspaces. 
• 3/7: 4.2 The null space and column space of a matrix. The kernel and image of a linear transformation.
• 3/9: 4.3 Linear independence and bases.
• 3/11: 4.4  Coordinates with respect to a basis.
• 3/14: 4.5 The dimension of a vector space.
• 3/16: 4.6 The rank of a matrix.
• 3/18: 4.7 Change of basis matrix.
• 3/28: Review
• 3/30: Exam #2 on 3.1-3.2, 4.1-4.7.
• 4/1: 5.1 Eigenvalues and eigenvectors.
• 4/4: 5.2   The characteristic polynomial.
• 4/6: 5.3 Diagonalization.
• 4/8: 5.4 Similarity.
• 4/11: 5.5 Complex eigenvalues of real matrices.
• 4/13: 6.1,6.7 Dot products, inner products, orthogonality, the orthogonal complement.
• 4/15: More 6.1 and 6.7.  Supplementary material on complex dot products (Hermitian inner products).
• 4/18: 6.2 Orthogonal and orthonormal sets.
• 4/20: 6.3  Orthogonal projection.
• 4/22: 6.4 The Gram-Schmidt process for finding orthogonal and orthonormal bases. The QR decomposition.
• 4/25: 6.5 Least squares solutions.
• 4/27: 6.8 (partially) Fourier series and Review.
  
• 4/29:  Exam #3 on sections 5.1-5.5, 6.1-6.5,6.7
• 5/2: 7.1 Symmetric (and Hermitian) matrices, orthogonal (and unitary) diagonalization.
• 5/4: 7.2 Quadratic forms.
• 5/6: 7.4 The singular value decomposition.
• 5/9: Review
• 5/11: Review
• 5/18: Final Exam 8:00-10:00  The final exam will cover the whole course with a slightly greater emphasis on Chapter 7.