Day 15: Linear Algebra

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Announcements:
- You have a homework set due today.
In chapter two we're thinking about vector spaces:
- Overview:
  A vector space is given by a collection of axioms (statements that one accepts, without question).
  A vector space is
  1. a set of vectors, subject to
  2. two operations: ${+,\cdot}$ (vector addition and scalar multiplication).
  with the following rules:
  1. Closure under both operations
  2. There is a zero vector in the set.
  3. Addition is commutative and associative, and additive inverses exist.
  4. The operations distribute one over the other.
- Section Two.I.2 concerns subspaces: Once you have a space, you may find that there are interesting subspaces -- e.g.
  1. the quadratic functions are an interesting subspace of the polynomials of degree three (where the coefficient of the term x³ can be 0).
  2. A plane passing through the origin is an interesting two-dimensional subspace of all of three-space.
  - Important fact:
    A subspace must contain the zero vector.
  - Important theorem:
    The span of a set of vectors in a vector space is a subspace.
- Section Two.II concerns Linear Independence:
  Mathematicians are always interested in doing as much as possible with the smallest possible.
  - Definition: a linear independent subset of vectors is one in which not one of the vectors in the set can be expressed using only the others.
Our first section for today is Vector Spaces.Basis and Dimension.Basis
- Definitions:
  1. Basis of vectors -- a sequence of vectors that
    1. is linearly independent, and
    2. spans the space.
    Note: our author is going a little off the beaten path here: generally a basis is just a subset, not a sequence (which is ordered).
    His definition is "necessary" for definition 1.13 (representation).
  2. Standard (natural, canonical) basis of ${\R^n}$ : the sequence of vectors ${\langle\overline{e}_1,\ldots,\overline{e}_n\rangle}$ with 1 in a single component, and zeros elsewhere.
  3. Representation: Given basis ${\langle\overline{\beta}_1,\ldots,\overline{\beta}_n\rangle}$ , a representation of ${\overline{v}}$ is given as a vector of the coefficients of the basis vectors: if ${\overline{v}=c_1\overline{\beta}_1+\ldots+c_n\overline{\beta}_n}$ , then the representation is the vector with components (or coordinates) ${(c_1,c_2,\ldots,c_n)}$ (where the order matters -- ah ha!).
- Theorems:
  1. A subset of vectors is a basis if and only if each vector in the space can be expressed as a unique linear combination of vectors in the subset.
- Examples:
  1. 1.16bc, p. 114
  2. 1.17a
  3. 1.18
  4. 1.26
Our second section for today is Vector Spaces.Basis and Dimension.Dimension
- Definitions:
  1. Finite-dimensional: A vector space is finite-dimensional if it has a basis with only finitely many vectors.
  2. Dimension of a vector space: the number of vectors in a basis (any basis!).
- Theorems:
  1. Exchange Lemma: we can replace a vector ${\overline{\beta}_i}$ in a basis by another vector ${\overline{v}}$ , if ${\overline{v}=c_1\overline{\beta}_1+c_2\overline{\beta}_2\cdots+c_i\overline{\beta}_i+\cdots+c_n\overline{\beta}_n}$ , with ${c_i\ne{0}}$ .
  2. Theorem: In any finite-dimensional vector space, all of the bases have the same number of elements (so we can read the dimension of the space off of a basis).
  3. Corollaries:
    1. No linearly independent set can have a size greater than the dimension of the enclosing space.
    2. Any linearly independent set can be expanded to make a basis.
    3. Any spanning set can be shrunk to a basis.
    4. In an n-dimensional space, a set of n vectors is linearly independent if and only if it spans the space.
- Examples:
  1. 2.15, p. 120
  2. 2.18b
  3. 2.19b
  4. 2.20

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