Date & time: Monday, March 17, 10:30-11:20

Place: Classroom

Coverage: Penney chapter 2, class notes: Linear Independence and Dimension

Be prepared to do problems and use knowledge from the following areas.

• 2.1: The test for linear independence
  • The criterion for (linear) independence of a set of vectors
  • A general technique for finding dependencies: Expressing "free elements" in terms of "pivot elements" See page 86.
  • The pivot columns of a matrix form a basis for the column space.
• Class notes: An infinite supply of vector spaces
  If A is a nonempty set and F is a field, then F^A, the set of all functions from A into F, forms a vector space under the "usual" definitions of function addition and multiplication of a function by a scalar. From Chapter 1, we know we can check whether a subset of a vector space is a subspace. A span is a subspace. So is any nonempty subset that is closed under the addition and scalar multiplication. Examples include
  • the set of all real-valued functions of a real variable and the following subspaces
  • the set of all polynomials
  • the set of all polynomials of degree at most n
  • the set of all continuous functions
  • the set of all differentiable functions
  • the set of all infinitely differentiable functions
• 2.2: Dimension
  • Definition of dimension: Dimension = number of vectors in a minimal spanning set.
  • Definition of basis
  • The standard basis for Rⁿ
  • Dimension theorem: dim(V) = n if V has a basis of n elements. ...
  • Theorem 1: If dim(V) = n, then any set of n+1 or more vectors must be linearly dependent. ...
  • Theorem 2: If dim(V) = n, then n given vectors are a basis if they either span or are linearly independent (for then the other property must necessarily hold, too).
• 2.3: Applications to systems
  • Row space
  • Non-zero Rows Theorem
  • Rank
  • "Row rank = column rank" theorem
  • Rank-Nullity Theorem
• Class notes: The Four Fundamental Subspaces and how to find a basis for each one
  • Row space. Basis: the nonzero rows of an echelon form
  • Column space. Basis: the pivot columns
  • Nullspace. Basis: the vectors appearing in the parametric form of the solution of the homogeneous system AX=0.
  • Left nullspace: the nullspace of the transpose
• Class notes: The Fundamental Theorem of Linear Algebra, Part I
  Let A by an m-by-n matrix. Let r = rank(A); r = # nonzero rows of any echelon form of A.
  • Row space = range of A^t; dimension = r.
  • Nullspace = {X|AX=0}; dimension = n-r.
  • Column space = range of A; dimension = r.
  • Left nullspace = nullspace of A^t; dimension = m-r.