Midterm II Review Sheet for Math 207, Spring 2007

Concepts to know

  1. what is a system of linear equations and the solution set thereof
  2. the coefficient matrix of a linear system
  3. the augmented matrix of a linear system
  4. what is a homogenous linear system and a trivial solution thereof
  5. what are elementary row operations
  6. what are elementary matrices
  7. what is a row-echelon matrix
  8. what is a reduced row-echelon matrix
  9. how a linear system can be inconsistent
  10. what is a matrix, column/row vector
  11. properties of matrix multiplication, addition and scalar multiplication
    1. when multiplication is defined, and what will then be the size of the product
    2. (lack of!) commutativity for matrix multiplication
    3. associativity, distributive law,
  12. what is a diagonal matrix? an upper/lower triangular matrix? a symmtric matrix
  13. what is the RRE form of an invertible (singular) matrix
  14. the inverse of a product of matrices
  15. the transpose of a product of matrices
  16. the inverse of a transpose
  17. the main theorem for an n×n matrix A on the equivalence of
    1. A is invertible
    2. the homogenous system with coefficient matrix A has only the trivial solution
    3. the RRE form of A is the identity matrix
    4. A is expressible as a product of elementary matrices
    5. A X = B is conistent for any B
    6. A X = B has exactly one solution for any B
  18. inverses and transposes of diagonal, upper and lower triangular matrices

Things to know how to do

  1. how to convert between a linear system and the correspoding matrices and/or the corresponding matrix equation
  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 multiply two matrices
  5. how to compute the trace of a matrix
  6. how to compute the transpose of a matrix
  7. how to compute the inverse of a matrix
    1. by the formula for the 2×2 case
    2. finding the inverse in general by row-reducing a matrix augmented with the inverse
  8. how to express an invertible matrix as a product of elementary matrices
  9. how to solve a linear system by inverting its coefficient matrix
  10. how to compute the matrix resulting from plugging a matrix into a polynomial

Possible extra problems

  1. §1.1: 1, 3, 4, 5, 12
  2. §1.2: 3, 4, 6, 8, 12, 19, 23, 24
  3. §1.3: 13, 27, 30, 32
  4. §1.4: 4, 7, 9, 11, 13, 20, 23
  5. §1.5: 1, 3, 6, 8, 10, 14, 24
  6. §1.6: 1, 4, 6, 9, 11, 13, 15, 19, 22
  7. §1.7: 1, 3, 5, 7, 11, 17, 19, 25


Jonathan Poritz (jonathan.poritz@gmail.com)