Day 26: Linear Algebra

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Today:

Announcements:
- Homework graded:
  1. 1.10, p. 164
  2. 1.13bc
  3. 1.30
- Your first project is due this coming Wednesday. Questions?
- Reminder: your next homework assignment is due Monday, 3/19 (special "hurry-up due date"):
  1. 2.8, p. 172
  2. 2.9
  3. 2.11
  4. 2.22
  5. 1.19, p. 179
  6. 1.20
  7. 1.23
  8. 1.25
  9. 1.27
- You have a new homework assignment due Friday. Hence you have something due every day next week. Ayeeeee!
In Chapter III we're thinking about mappings: we're looking at homomorphisms between spaces (rather than isomorphisms), aka "linear maps", which preserve the structure of the two operations across the spaces.
- Maps Between Spaces.Homomorphisms.Rangespace and Nullspace
  - Overview:
    In this section we discover that a homomorphism ${h:V\to{W}}$ induces ("generates") two special subspaces: one in the domain space, V, and one in the "codomain" space W.
    The important new space generated in the domain is the space of vectors that get mapped to the zero vector of space W (vectors that are "annihilated" by the homomorphism -- that sounds pretty exciting!).
    The important new space generated in the codomain W is the space of vectors that are images of vectors of space V.
  - Definitions:
    1. rangespace: The rangespace of a homomorphism ${h:V\to{W}}$ is ${\mathcal{R}(h)=\{h(\overline{v})|\overline{v}\in{V}}}$ sometimes denoted ${h(V)}$ . The dimension of the rangespace is the map's rank. (See example 2.5, p. 183.)
    2. nullspace: The nullspace (or kernel) of a linear map ${h:V\to{W}}$ is the inverse image of ${\overline{0}_W}$ : ${\mathcal{N}(h)=\{\overline{v}\in{V}|h(\overline{v})=\overline{0}_W}}$ . The dimension of the nullspace is the map's nullity.
    3. nonsingular: A linear map that is one-to-one is nonsingular.
  - Theorems:
    1. 2.1 Lemma: Under a homomorphism, the image of any subspace of the domain is a subspace of the codomain. In particular, the image of the entire domain space, which is the range of the homomorphism, is a subspace of the codomain. [Hence a homomorphism generates lots of subspace!]
    2. 2.14 Theorem: A linear map's rank plus its nullity equals the dimension of its domain vector space V. [Everything goes somewhere, essentially.]
    3. 2.17 Corollary: The rank of a linear map is less than or equal to the dimension of the domain. Equality holds if and only if the nullity of the map is zero. [In this case the map is one-to-one.]
    4. 2.18 Lemma: Under a linear map, the image of a linearly dependent set is linearly dependent.
    5. 2.21 Theorem:
  - Examples:
    1. 2.22, p. 190
    2. 2.23
    3. 2.24 (let's also give example maps)
- Maps Between Spaces.Computing Linear Maps.Representing Linear Maps with Matrices
  "...a linear map is determined by its action on a basis."
  - Summary: A homomorphism is determined by its action on a basis. That is, if ${\langle\overline{\beta}_1,\ldots,\overline{\beta}_n\rangle}$ is a basis of a vector space V and ${\overline{w}_1,\ldots,\overline{w}_n}$ are (perhaps not distinct) elements of a vector space W, then there exists a homomorphism from V to W sending ${\overline{\beta}_i}$ to ${\overline{w}_i}$ , and that homomorphism is unique.
    Let's take a good look at example 1.1 from this section, p. 193.
    At the end of this example, bottom of p. 194, the author defines (essentially creates a) matrix multiplication. This definition is crucial for all that follows.
  - Definitions:
    1. matrix representation of homomorphism h with respect to B, D (bases of domain and codomain, respectively:
    2. matrix multiplication:
      
      Observe the essential alignment of dimensions:
      - the number of rows in the vector multiplied by the matrix must conform to the number of columns of the matrix.
      - The number of rows of the result will conform to the number of rows of the matrix.
      "A good way to view a matrix-vector product is as the dot products of the rows of the matrix with the column vector."
      
      "Matrix-vector product can also be viewed column-by-column."
  - Theorems:
    1. Theorem 1.4:
  - Examples:
    1. Really important example: 1.8, p. 198
    2. 1.11, p. 200 -- let's think of the multiplication in as many different ways as possible...
    3. 1.13 -- flashback to linear systems.

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