| dc.description.abstract | You are about to teach a course that will probably give students their second
exposure to linear algebra. During their first brush with the subject, your
students probably worked with Euclidean spaces and matrices. In contrast,
this course will emphasize abstract vector spaces and linear maps.
The audacious title of this book deserves an explanation. Almost all
linear algebra books use determinants to prove that every linear operator on
a finite-dimensional complex vector space has an eigenvalue. Determinants
are difficult, nonintuitive, and often defined without motivation. To prove the
theorem about existence of eigenvalues on complex vector spaces, most books
must define determinants, prove that a linear map is not invertible if and only
if its determinant equals 0, and then define the characteristic polynomial. This
tortuous (torturous?) path gives students little feeling for why eigenvalues
exist.
In contrast, the simple determinant-free proofs presented here (for example,
see 5.21) offer more insight. Once determinants have been banished to the
end of the book, a new route opens to the main goal of linear algebra—
understanding the structure of linear operators.
This book starts at the beginning of the subject, with no prerequisites
other than the usual demand for suitable mathematical maturity. Even if your
students have already seen some of the material in the first few chapters, they
may be unaccustomed to working exercises of the type presented here, most
of which require an understanding of proofs. | en_US |
