## Linear Algebra

Spring 2018

Syllabus.

The course so far.

Record of assignments.

Solutions distributed in class.

The Greek alphabet.

Some tips on dealing with proofs.

A proof from properties.

Dot products and the Cauchy-Schwarz Inequality.

Cauchy-Schwarz and Triangle Inequalities (Calc III handout).

Proof of Theorem 1.13 (p. 54).

A proof that matrix multiplication is associative.

Some basic facts about inequalities.

Equivalence relations and partitions.

A proof of Lemma 2.7 (p118).

Classwork: Matrix mult and linear combinations.

p110#13b (ec problem).

Expressing solution sets using linear combinations.

Some consequences of the invariance of the Laplace expansion.

The cross product.

The proof of Theorem 3.7 [det(AB)=det(A)det(B)] (p66).

Classical adjoints and inverses.

One way in which section 3.4 can be applied.

Classwork on eigenvalues and eigenspaces.

Definition of vector space and a page from your calculus text.

The first theorem about abstract vector spaces.

An expanded version of Theorem 4.5 (p. 250).

From "dependent" to "independendent" via MAT 225.

A useful test for linear dependence/independence.

Why the r.r.e.f. is unique.

Some characterizations of linear dependence/independence.

A proof of Theorem 3. 7 [det(AB)=det(A)det(B)]using elementary matrices.

A program to compute determinant by row reduction.

Extra Credit Project (formula for Fibonacci numbers).

Proof of Lemma 4.11 (slightly reworked).

Outline of Sec. 4.5 of the text.

Outline of Sec. 4.6 of the text.

Outline of Sec. 4.7 of the text.

An introduction to linear functions.

A proof that rref is unique.

The transition matrix between two coordinatizations.

The matrix of a rotation in the plane.

Proof of Theorem 5.8 and explanation why matrix mult. is associative.

A good chapter on induction (from a discrete math text).