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.
An alternate proof of Theorem 1.10.
Cauchy-Schwarz and Triangle Inequalities (Calc III handout).
Example: an inductive proof that requires a double basis case.
Proof of Theorem 1.15 (p. 60).
A proof that matrix multiplication is associative.
A more detailed proof that matrix multiplication is associative.
Row operations through matrix multiplication.
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).
A matrix-based proof of Lemma 2.8.
Expressing solution sets using linear combinations.
One determinant calculated six ways.
Some consequences of the invariance of the Laplace expansion.
The cross product.
The proof of Theorem 3.7 [det(AB)=det(A)det(B)] (p177).
Classical adjoints and inverses.
Classical adjoints.
One way in which section 3.4 can be applied.
Notes on similar matrices.
Classwork on eigenvalues and eigenspaces.
Mathematical abstraction in the modern sense.
Definition of vector space and a page from your calculus text.
The first theorem about abstract vector spaces.
The intersection of subspaces is a subspace.
An expanded version of Theorem 4.5 (p. 245).
Linear independence versus linear dependence.
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 4.10.
Outline of Sec. 4.5 of the text.
Outline of Sec. 4.6 of the text.
The transition matrix.
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).