Date & time: Friday, October 17, 2:30-3:20

Place: Classroom

Coverage: Schneider & Barker, chapter 3 to p. 152; class notes; section 5.1: Vector spaces, basics of linear transformations

Be prepared to do problems and answer basic questions testing your knowledge the following topics.

• 3.1: Vectors and vector spaces
  • Definition of vector space
  • Examples 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.
  • Problem: Identifying vector space properties used in proofs
• 3.2: Subspaces and linear combinations
  • Subspaces: definition & criterion, Theorem (3.2.2)
  • Linear combinations
  • "Span," "spans"
• 3.3: Linear dependence and linear independence
  • Definitions
  • A general technique for finding dependencies of column vectors: Let A be the matrix of column vectors; solve Ax = 0 by row reducing A to echelon form.
• 3.4: Bases
  • Definition of basis: A basis must span and be linearly independent.
  • Definition of dimension: Dimension = number of vectors in a basis.
  • Dimension theorem: If V has a basis of n elements, then any other basis has n elements. Thus "dimension" is well defined in finite-n cases.
  • Proposition (3.4.9): If dim V = n, then n is an upper bound on the number of vectors in any linearly independent family and a lower bound on the number of vectors in any spanning family.
  • Theorem (3.4.10): If dim V = n, then n given vectors are a basis iff they either span or are linearly independent (for then the other property must necessarily hold, too).
• 3.5: Bases and representations
  • The representation of a vector z with respect to a basis (v¹, ..., vⁿ) is the vector [c₁, ..., c_n]^T of "coordinates" (scalars) for which z = c₁ v¹ + ... + c_n vⁿ.
  • Theorem (3.5.2): Two bases "X" and "Y" are related by a nonsingular matrix P--"Y" = "X" P--and then (Theorem (3.5.5)) the representation of a vector z w.r.t. "Y" is P^-1 times the representation of z w.r.t. "X".
• The Four Fundamental Subspaces and how to find a basis for each one
  • Row space. Basis: the nonzero rows of an echelon form
    3.6: Row space of a matrix
    • Main theorem (3.6.10): Logical equivalents of row equivalence
  • Column space. Basis: the pivot columns
    3.7: Column equivalence
    • The definitions and theorems are the "duals" of those for rows.
  • Nullspace. Basis: the vectors appearing in the parametric form of the solution of the homogeneous system Ax=0. (Use the "free" variables as the parameters.)
  • Left nullspace: the nullspace of the transpose
• Class notes: The Fundamental Theorem of Linear Algebra, Part I
  Part I of this theorem tells us the dimensions of the four fundamental subspaces of a matrix. It incorporates these important theorems: row rank = column rank (= rank); rank+nullity = dimension.
  Let A by an m-by-n matrix. Let r = rank(A); r = # nonzero rows of any echelon form of A.
  • Row space = span of rows of A; dimension = r.
  • Null space = {x|Ax=0}; dimension = n-r.
  • Column space = range of A; dimension = r.
  • Left nullspace = nullspace of A^T; dimension = m-r.
• 5.1: Linear transformations: Definitions
  • Definition and equivalent characterization of "linear transformation"
  • Addition, scalar multiplication, and composition (looks like multiplication) of linear transformations
  • T(x) = Ax where A is an m-by-n matrix and x is an n-by-1 matrix (vector) defines a linear transformation.