Day 24: Linear Algebra

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Announcements:
- Your tests returned:
  1. The exam
  2. The key
Do you have any questions about the material of Chapter II?
In chapter three we're thinking about mappings:
- Maps Between Spaces.Isomorphisms.Definition and Examples
  - Definitions:
    1. isomorphism: An isomorphism between two vector spaces V and W is a map that
      1. is one-to-one and onto from one set of vectors to the other; and
      2. preserves structure (the two operations): if ${f(\overline{v}_1+\overline{v}_2)=f(\overline{v}_1)+f(\overline{v}_2)$ and if ${f(r\cdot\overline{v}_1)=r\cdot{f}(\overline{v}_1)$ where it is important to realize that the two operations ${+,\cdot}$ of addition and multiplication by a scalar are generally different in the two spaces.
    2. automorphism: an isomorphism between a space and itself.
  - Theorems:
    1. 1.8 Lemma: An isomorphism maps the zero vector of V to the zero vector of W.
    2. 1.9 Lemma: For any map ${f:V\rightarrow{W}}$ betweeen vector spaces the following checks that the operations are preserved:
      f preserves linear combinations:
      ${f(c_1\overline{v}_1+c_2\overline{v}_2)=c_1f(\overline{v}_1)+c_2f(\overline{v}_2)$
    These two lemmas help us in two ways:
    1. The first gives us a quick check on a potential isomorphism: does it map one zero vector to the other zero vector? If not, it's not an isomorphism.
    2. The second gives us a check for the preservation of operations: just check that linear combinations are preserved (the second part above).
  - Examples:
    1. Is it obvious that the space of points in the Cartesian plane ${M=\{(x,y)|x,y\in\R\}}$ with the usual operations is isomorphic to the space of points ${N=\{(y,x)|x,y\in\R\}}$ with the usual operations?
      Give an example mapping. That is, explained how vectors are carried into vectors, preserving operations.
      This is an example of an automorphism.
    2. 1.11, p. 164
    3. 1.13ad
    4. 1.27
- Maps Between Spaces.Isomorphisms.Dimension Characterizes Isomorphism
  - Theorems:
    1. Isomorphism is an equivalence relation between vector spaces.
    2. Vector spaces are isomorphic if and only if they have the same dimension.
    3. Corollary: A finite-dimensional vector space is isomorphic to one and only one of the vector spaces ${\R^n}$ .
  - Examples:
    1. 2.8-2.13, p. 172
- Maps Between Spaces.Homomorphisms.Definition
  - Definitions:
    1. homomorphism: A function between vector spaces ${h: V\rightarrow{W}}$ that preserves the operations of addition ${h(\overline{v}_1+\overline{v}_2)=h(\overline{v}_1)+h(\overline{v}_2)$ and scalar multiplication: ${h(r\cdot\overline{v}_1)=r\cdot{h}(\overline{v}_1)$ is a homomorphism or linear map.
    2. linear transformation: A linear map (homomorphism) from a space into itself: ${t:V\rightarrow{V}}$ is a linear transformation.
      An automorphism is an example of a linear transformation.
  - Theorems:
    1. 1.6 Lemma: A homomorphism sends a zero vector to a zero vector.
    2. 1.7 Lemma: Each of these is a necessary and sufficient condition for to be a homomorphism:
      1. ${f(c_1\overline{v}_1+c_2\overline{v}_2)=c_1f(\overline{v}_1)+c_2f(\overline{v}_2)$
      2. ${f(c_1\overline{v}_1+\ldots+c_n\overline{v}_n)=c_1f(\overline{v}_1)+\ldots+c_nf(\overline{v}_n)$
    3. A homomorphism is determined by its action on a basis. That is, if ${\langle\beta_1,\ldots,\beta_n\rangle}$ 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 ${\beta_i}$ to ${\overline{w}_i}$ , and that homomorphism is unique.
    4. For vector spaces V and W, the set of linear functions from V to W is itself a vector space, a subspace of the space of all functions from V to W. It is denoted ${L(V,W )}$ .
      This is truly something to think about, to reflect on: we have a new vector space, whose vectors are mappings from one vector space to another. We have to define how to combine mappings (sums and scalar products), but -- Whoa!
  - Examples:
    1. 1.17, p. 179
    2. 1.18
    3. 1.27

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