Math 311-101 - Test I Review

General Information

Topics Covered

Systems & matrices

• Linear sytems
  • Augmented matrix form. Convert a system to and from augmented matrix form.
  • Row operations and equivalent systems. Know the three types of row operations and that they result in an equivalent system.
  • Row echelon form of a matrix. Be able to use Gauss elimination to put a matrix in row echelon form. Know what lead variables and free variables are.
  • Reduced row echelon form. Be able to use Gauss-Jordan reduction to put a matrix in this form. This makes the connection between lead (dependent) variables and free variables explicit.
  • Special types of systems: homogeneous, overdetermined, underdetermined.
• Matrices
  • Matrix algebra. Sum, product, scalar multiples, row vectors, column vectors, transpose, symmetric matrix, identity matrix, zero matrix, size of a matrix, (i,j) entry, notation.
  • Inverse of a matrix. Know how the inverse is defined. Also, know the terms invertible, nonsingular, and singular. Be able to be find the inverse of a matrix or show that a matrix is singular via row reducing (A|I)
  • Application to networks and graphs. Adjacency matrix A. Interpretation of entries in A^k.
• Elementary matrices
  • Three types of elementary matrices and correspondence to row operations.
  • Be able to show that if A and B are invertible, then AB is invertioble and (AB)^-1 = B^-1A^-1
  • Definition of row equivalence of matrices.
  • Equivalent conditions:
    1. A is nonsingular.
    2. Ax = 0 has only x = 0 as a solution.
    3. A is row equivalent to I.
  • LU factorization. Upper triangular, lower triangular and diagonal matrices. Be able to factor a matrix into LU form.

Determinants

• Basic properties. Know the basic properties given in my notes on determinants.
  • Determinants of special matrices. The determinant of an upper triangular, lower triangular, or diagonal matrix is the product of the diagonal entries.
  • Row operations. Be able to use row operations to find a determinant.
  • Elementary matrices. Know the determinant of the three types of elementary matrices.
  • Inverses. Be able to determine whether an n×n matrix A is invertible from knowing det A.

Review of 3D Geometry

• The dot and cross product. Be able to find the areas of triangles and parallelograms and volumes of parallelepipeds using vector methods.
• Lines, planes and distances. Be able to find the parametric forms for the equations of lines and planes, the coordinate form for the equation of a plane, and distances between various geometric objects - e.g., point and plane, two skew lines, etc.

Vector spaces

• Basic ideas. Addition, multiplication by scalars, and being closed under addition and scalar multiplication. Notation for special spaces: Rⁿ, R^m×n, P_n, C[a,b], C^k[a,b].
• Subspaces
  • Know the definition of a subspace.
  • Know the test to determine whether a subset S of a vector space V is a subspace: (i) Is 0 in S? (ii) Is S closed under + ? (iii) Is S closed under · ? Be able to use it to determine whether subsets are subspaces.
• Spans. Linear combinations. Spanning sets. Null space of a matrix, N(A).
• Linear independence, linear dependence. Know the definitions of these. Be able to test a set of vectors to determine wheteher it is LI or LD.
• Basis and dimension. Be able to define these terms, along with the term dimension. Be able to show that if a vector is in the span a set of linearly independent vectors, then there is only one way to write it as a linear combination of the LI vectors.
• Coordinates. Be able to find coordinates of a vector relative to a given (ordered) basis.