Midterm II Review Sheet for Math 207, Spring 2009

Concepts to know

1. what is a system of linear equations and the solution set thereof
2. the structures of possible solution sets of linear systems — empty, a single point, or infinite (and the geometric picture corresponding to each situation)
3. the coefficient matrix of a linear system
4. the augmented matrix of a linear system
5. what is a homogenous linear system
6. what are elementary row operations
7. what is a row-echelon matrix [REF]
8. what is a reduced row-echelon matrix [RREF]
9. what free variables in the solution set of a linear system correspond to in the RREF matrix of the system
10. the rank of a matrix
11. how a linear system can be inconsistent
12. what inconsistency of a linear system corresponds to in the RREF version of the augmented matrix of the system
13. the span of a set of vectors
14. the theorem that a linear system is consistent if and only if its right hand side, made into a vector, is in the span of the columns of the coefficient matrix
15. linear dependence/independence of a set of vectors

Things to know how to do

1. how to convert between a linear system and the correspoding coefficient and augmented matrices
2. how to row-reduce a matrix
   1. Gaussian elimination
   2. Gauss-Jordan elimination
3. how to find the solution set of a linear system whose matrix is in reduced row-echelon form.
4. how to test a set of vectors for linear dependence
5. certain kinds of applications:
   1. flows in networks (e.g., traffic on roads, current in circuits [i.e. Kirchhoff's laws])
   2. word problems about various quantities of different substances, such as agricultural products
   3. finding polynomials which pass through given points
   in each case, setting up and solving the linear system...