These are notes for a course in linear algebra. Students taking this course at Millersville University are assumed to have had, or be currently enrolled in, Calculus 3. Our Calculus 3 course covers vectors in 3 dimensions, including dot and cross products. Hence, my notes don't spend much time on basic vector arithmetic, since it would be redundant for our students.

There are several things about these linear algebra notes that are a little unusual. One novelty is that I've done as much linear algebra as possible over fields of nonzero characteristic. In practice, I confine myself to examples over the integers mod 2, 3, and 5, but I think this is enough to get the point across. To this end, the first section discusses commutative rings with identity and fields.

(This isn't a big deal. A typical proof of something in linear
algebra where the ground ring is a commutative ring with identity rather
than a field uses algebraic facts like commutativity of addition that
everyone takes for granted. So if you just *do* the proof without
belaboring the fact that the ground ring isn't a field, it doesn't cause
any confusion or puzzlement.)

Determinants are defined axiomatically, over a commutative ring with
identity, and I think I've given complete proofs of all the results.
The axiomatic presentation makes sense to me (probably because that
was the way I learned in, from Hoffman and Kunze as an undergrad). It's
a good opportunity to show students how an object can be defined
by *properties*. Then you show it exists by construction, and
finally you show it's unique.

Doing determinants over a commutative ring with identity is actually no harder than doing determinants over a field: The algebraic manipulations in the proofs rarely make any reference to the number system you're working in. But it's also a matter of intellectual honesty, since you will later use determinants with entries in a polynomial ring to compute eigenvalues.

Finally, the algorithm for the computation of the exponential of a matrix uses an approach which I believe is due to Richard Williamson. Of the approaches I've seen, it seems to be the simplest for hand computation, and (modulo computing the eigenvalues, which is always an issue), can be implemented in computer algebra systems with little difficulty.

- Rings and fields

[PDF file] - Matrix arithmetic by example

[PDF file] -
Properties of matrix arithmetic (with proofs)

[PDF file] - Row reduction

[PDF file] - Inverses and elementary
matrices

[PDF file] - Axioms for determinant
functions; existence of a determinant function (defined by
cofactor expansion); the effect of row operations

[PDF file] - Uniqueness of determinant
functions; the permutation representation; properties of
determinants (multiplicativity, |A| = |A^T|); Cramer's rule;
the adjoint formula for an inverse

[PDF file] -
Vector spaces and subspaces

[PDF file] - Linear independence

[PDF file] - Spanning sets

[PDF file] - Bases for vector space

[PDF file] - Row space, column space, null space;
the rank of a matrix

[PDF file] - Linear transformations

[PDF file] - Change of basis

[PDF file] -
Matrices and change of basis for linear transformations

[PDF file] - Eigenvalues and
eigenvectors

[PDF file] - The Cayley-Hamilton
Theorem

[PDF file] -
Applications to constant coefficient homogeneous differential equations

[PDF file] - Solving systems of linear
differential equations using eigenvectors

[PDF file] - The
exponential of a matrix

[PDF file] - Inner product spaces

[PDF file] - Coordinate transformations
in the plane (Translations, rotations, reflections)

[PDF file] - Unitary and Hermitian matrices

[PDF file] -
The Spectral Theorem and the Principal Axis Theorem

[PDF file] - An introduction to Fourier series

[PDF file]

Copyright 2010 by Bruce Ikenaga