Day 10: Linear Algebra

• 1.1bd, p. 38
• 1.4
• 1.8
  • Sorry: this was not stated too well. It was very confusing.
  • I want you to practice drawing vectors into the coordinate system.
  • Don't draw illegal axes.

• 2.10be
• 2.18
• 2.26

• The process of Gauss reduction is not-unique. However we have seen that it preserves the solution set of a linear system. Hence, every reduced augmented matrix system represents an equivalent system, from the standpoint of the solutions.

• This section shows us how to reduce to a single elementary form (Gauss-Jordan form) that exposes the solution in a nice way.

• Motivation: actually the first time Zach got up to solve a system, he reduced it all the way to Gauss-Jordan form -- to a diagonal matrix, containing just ones on the diagonal, and we noticed then that we could just read off the solution that way. That's what we're doing in this section -- just pushing on with our row-reductions until we get it in "Zach-form".

• Gauss-Jordan reduction

  the matrix has only ones for leading entries in rows, and each leading one is alone in its column.
  In a way, it "represents" its column.

• Gauss-Jordan reduction leads to an augmented system in reduced echelon form:
  • The solution can be just read off...
  • We can also see the dimension of the solution easily (back to Jenny's question regarding 2.29, p. 19) -- the example I gave was an example in Gauss-Jordan form.
  • But it requires more (and often unnecessary) arithmetic -- sometimes you just want to backsolve ASAP, and not sit around admiring your solution's "exposure"....

• row equivalent matrices -- interchangeable by elementary row operations
• row equivalence classes.

• Elementary row operations are reversible.
• Between matrices, "reduces to" is an equivalence relation.

  An equivalence relation is one such that

  1. A is equivalent to itself (reflective)
  2. A is equivalent to B iff B is equivalent to A (symmetric)
  3. If A is equivalent to B and B is equivalent to C, then A is equivalent to C (transitive)

• (Linear Combination Lemma) A linear combination of linear combinations is a linear combination.

• Where one matrix reduces to another, each row of the second is a linear combination of the rows of the first.

• In an echelon form matrix, no nonzero row is a linear combination of the other rows.

• If two echelon form matrices are row equivalent then the leading entries in their first rows lie in the same column. The same is true of all the nonzero rows -- the leading entries in their second rows lie in the same column, etc.

• Here's the big theorem: Each matrix is row equivalent to a unique reduced echelon form matrix.

• 1.7c, p. 51
• 1.8c
• 1.10
• 1.11b
• 2.10cd, p. 59
• 2.12a
• 2.16
• 2.18b
• 2.23

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: (vector addition and scalar multiplication).

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. ,
   2. ,

8. 
9. 

• 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

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.