Day 21: Linear Algebra

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Today:

Announcements:
- Your homework (that I didn't get back to you last time):
  - 2.16, p. 120
  - 2.18ac
  - 2.23:
    "2.9 Corollary: No linearly independent set can have a size greater than the dimension of the enclosing space." (p. 119)
    "2.13 Corollary: In an n-dimensional space, a set of n vectors is linearly independent if and only if it spans the space." (p. 119)
    Most of you showed that the four vectors were independent (although many of you didn't actually tell me that); Alex showed that the four vectors spanned the space. The corollary says that either works to show that n vectors is a basis.
- You have an assignment due today.
- You do have a test Friday, over Chapter II.
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.
      We write , read "V is isomorphic to W", when such a map exists.
    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 betweeen vector spaces the following are equivalent:
      1. f preserves structure (the two operations): ${f(\overline{v}_1+\overline{v}_2)=f(\overline{v}_1)+f(\overline{v}_2)$ and ${f(r\overline{v}_1)=rf(\overline{v}_1)$
      2. f preserves linear combinations: ${f(c_1\overline{v}_1+c_2\overline{v}_2)=c_1f(\overline{v}_1)+c_2f(\overline{v}_2)$
      3. f preserves linear combinations of any finite number of vectors: ${f(c_1\overline{v}_1+\ldots+c_n\overline{v}_n)=c_1f(\overline{v}_1)+\ldots+c_nf(\overline{v}_n)$
    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
    5. 1.21
- 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

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