Page 9 - Linear Algebra Done Right

Preface to the Instructor

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 op-
erator on a finite-dimensional complex vector space has an eigenvalue.
Determinants are difficult, nonintuitive, and often defined without mo-
tivation. To prove the theorem about existence of eigenvalues on com-
plex 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.
In contrast, the simple determinant-free proofs presented here of-
fer 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 prerequi-
sites 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.

• Vector spaces are defined in Chapter 1, and their basic properties
are developed.

• Linear independence, span, basis, and dimension are defined in
Chapter 2, which presents the basic theory of finite-dimensional
vector spaces.

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