Suggested reading: Sections 1.3, 1.4, 1.5, and 1.7.
Homework Assignment #3. Due Wednesday, September 20.
The solutions to Quiz 1 are posted above. The quiz itself is graded out of 10 points, but the total score is out of 12 points: if you turned in your homework you receive an additional two points.
Monday's class: Introduced the span of a collection of vectors. Defined the dot product and went over basic properties. Defined the product of an (m x n )-matrix and a vector in R^n in terms of linear combinations. Explained how to compute this product using dot products. Introduced the coefficient matrix and stated a theorem establishing the equivalence of solution sets between the representation of a linear system as an augmented matrix, a vector equation, and a matrix equation. Defined the column space of a matrix and went over a theorem establishing the equivalence of several conditions, namely the consistency of a matrix equation, the column space of a matrix being as large as possible, and the existence of pivot positions in every row of a matrix.
Wednesday's class: Introduced homogeneous linear systems. Showed, via a worked example, how to express the general solution of a homogeneous linear system as a span of vectors. Proved a theorem (Theorem 6) relating the general solution of Ax=b to the general solution of Ax=0. Defined linear independence for a set of vectors. Made basic observations of when a set with one or two vectors is linearly independent.