Day 13: Linear Algebra

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A picture of homework's value:

The regression analysis:

Linear Regression:        Estimate            SE              Prob

Constant                 50.5826        (7.77127)           0.00000
Variable 0               5.28692        (1.69215)           0.00558

R Squared:               0.339400
Sigma hat:               12.9886
Number of cases:                21
Degrees of freedom:             19

In Chapter Two, we study a very general 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.
Section Two.I.2 concerns subspaces: Once you have a space, you may find that there are interesting subspaces -- e.g. the quadratic functions are an interesting subspace of the polynomials of degree three (where the coefficient of the term x³ can be 0).
In three-space, we can think of a plane through the origin as an interesting subspace.

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 S of vectors: [S], 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
Section Two.II concerns Linear Independence
Mathematicians are always interested in doing as much as possible with the smallest possible. For example, we didn't need to have an axiom that

${0\cdot\overline{v}=\overline{0}}$
since we can deduce it from our other axioms.
In this section, we study the problem of finding a "minimal spanning set" -- just enough to do the job of generating everything in a space.

Vector Spaces.Linear Independence.Definition and Examples
- Definitions:
  1. linear independent subset of vectors -- not one of the vectors in the set can be expressed using only the others.
  3. Span of a set of vectors: the set of all linear combinations of the set of vectors.
- Theorems:
  1. Any finite subset has a linearly independent subset with the same span.
  2. Any subset of a linearly independent set is also linearly independent. Any superset of a linearly dependent set is also linearly dependent.
- Examples:
  1. 1.18a, p. 106
  2. 1.19a
  3. 1.20a
  4. 1.21a
  5. 1.37

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