Midterm II Review Sheet for Math 207, Fall 2011

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 Rank Theorem
  14. the span of a set of vectors
  15. 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
  16. linear [in]dependence of a set of vectors
  17. matrix operations, such as addition, subtraction, scalar multiplication and matrix multiplication
  18. properties of matrix operations, such as associativity and the distributive law — but not commutativity
  19. the inverse of a matrix, and when a matrix is invertible
  20. examples of invertible matrices and their inverses, and of non-invertible matrices
  21. inverses for 2×2 matrices — when they exist, a formula which computes them

Things to know how to do

  1. convert between a linear system and the correspoding coefficient and augmented matrices
  2. row-reduce a matrix
    1. Gaussian elimination
    2. Gauss-Jordan elimination
  3. find the solution set of a linear system whose matrix is in reduced row-echelon form.
  4. test a set of vectors for linear dependence
  5. test if a vector is in the span of a set of other vectors
  6. 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
    in each case, setting up and solving the linear system...
  7. manipulate matrices (and vectors) using matrix operations
  8. write down the inverse of a 2×2 matrix with the formula
  9. find the inverse of a general matrix with the Gauss-Jordan method
  10. use the inverse of the coefficient matrix to find the solution of a linear system