Day 18: Linear Algebra

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Today:

Announcements:
- Your homework from last time:
  - 1.17bc, p. 114
  - 1.21: notation
    - You're asked for a basis. Just a handful of vectors.
    - Don't give a span.
  - A serious concern: 1.27
    - When is mathematical rigor important?
    - When is notation (cryptic or otherwise) important?
      - What is the definition of a basis?
    - How long does notation remain cryptic?
    - I don't believe that our author is given to gratuitous notation....
  - May I object to cryptic answers? State
    - implications: ${\longrightarrow}$
    - conclusions: ${\therefore}$
- You have a homework set due today (p. 114)
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.
Then there were bases:

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.
  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.
Then there is the dimension, which we just touched on last time:

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. 2.19b, p. 120
Now we want to take a look at linear systems from the vector space perspective:

Vector Spaces.Basis and Dimension.Vector Spaces and Linear Systems
- Definitions:
  1. row space: The row space of a matrix is the span of the set of its rows. The row rank is the dimension of the row space, the number of linearly independent rows.
  2. column space: The column space of a matrix is the span of the set of its columns. The column rank is the dimension of the column space, the number of linearly independent columns.
  3. transpose: The transpose of a matrix is the result of interchanging the rows and columns of that matrix. That is, column j of the matrix A is row j of A^trans , and vice versa.
  4. rank: The rank of a matrix is its row rank or column rank (they're the same!).
- Theorems:
  1. Lemma: row-equivalent matrices have the same row space, and hence also, the same row rank.
  2. Lemma: The nonzero rows of an echelon form matrix make up a linearly independent set.
  3. Lemma: Row operations do not change the column rank.
  4. Theorem: The row rank and column rank of a matrix are equal.
  5. Theorem: For linear systems with n unknowns and with matrix of coefficients A, the statements
    1. the rank of A is r
    2. the space of solutions of the associated homogeneous system has dimension n-r
    are equivalent.
  6. Corollary (important! It's kind of buried in the text on p. 126): For the matrix A_nxn, the statements
    1. the rank of A is n
    2. A is non-singular
    3. the rows of A form a linearly independent set
    4. the columns of A form a linearly independent set
    5. any linear system whose matrix of coefficients is A has one and only one solution
    are equivalent.
- Examples:
  1. 3.16ad, p. 127
  2. 3.17a
  4. 3.26
  5. 3.33
  6. 3.36

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