You are probably about to teach a course that will 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 must exist.
