Day 12: Linear Algebra

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In Chapter Two, we begin to study a very general, abstract notion: a vector space.

Vector Spaces.Definition of Vector Space.Definition and Examples
Our author makes a compelling argument that studying general things is sometimes better than studying specific things. And so we are going to study this general notion, the vector space.
- What is a vector space (over )?
  A vector space is given by a collection of axioms: statements that one accepts, without question. First of all, a vector space is
  1. a set of vectors, subject to
  2. two operations: ${+,\cdot}$ (vector addition and scalar multiplication).
  Now here are the rules:
  1. The set of vectors is closed under addition: that is, the sum of vectors from the set is again a vector in the set.
  2. Addition of vectors is commutative.
  3. Addition of vectors is associative.
  4. There is a zero vector in the set.
  5. There are additive inverses for each vector.
  6. The set of vectors is closed under scalar multiplication.
  7. Distributive laws apply:
    1. ${(r+s)\cdot\overline{v}=r\cdot\overline{v}+s\cdot\overline{v}}$ ,
    2. ${r\cdot(\overline{u}+\overline{v})=r\cdot\overline{u}+r\cdot\overline{v}}$ ,
  8. ${(rs)\cdot\overline{v}=r\cdot(s\cdot\overline{v})}$
  9. ${1\cdot\overline{v}=\overline{v}}$
- Now we want to look at some examples of vector spaces.
  - vectors! (in the sense that we've been studying vectors so far). Example 1.3, p. 78 verifies this.
  - functions
  - trivial vector space (with just one vector, the zero vector)
  - 2x2 matrices
- Examples:
  1. 1.18
  2. 1.19
  3. 1.20
  4. One lemma is proven in this section, and it's worth looking at: 1.17, p. 85.
The next section is Two.I.2:

Vector Spaces.Definition of Vector Space.Subspaces and Spanning Sets
- Definitions:
  1. Subspace: a subset of V that is also a vector space, using the inherited operations.
  2. Proper and improper subspaces
  3. Span of a set of vectors: the set of all linear combinations of the set of vectors.
- Theorems:
  1. A subset S of vectors in a vector space is a subspace if it is closed under linear combinations.
  2. The span of a set of vectors in a vector space is a subspace.
- Examples:
  1. 2.20ac, p. 95
  2. 2.22ab
  3. 2.25ab
  4. 2.26c

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