Page 10 - Linear Algebra Done Right

Preface to the Instructor
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• Linear maps are introduced in Chapter 3. The key result here
is that for a linear map T, the dimension of the null space of T
plus the dimension of the range of T equals the dimension of the
domain of T.
• The part of the theory of polynomials that will be needed to un-
derstand linear operators is presented in Chapter 4. If you take
class time going through the proofs in this chapter (which con-
tains no linear algebra), then you probably will not have time to
cover some important aspects of linear algebra. Your students
will already be familiar with the theorems about polynomials in
this chapter, so you can ask them to read the statements of the
results but not the proofs. The curious students will read some
of the proofs anyway, which is why they are included in the text.

• The idea of studying a linear operator by restricting it to small
subspaces leads in Chapter 5 to eigenvectors. The highlight of the
chapter is a simple proof that on complex vector spaces, eigenval-
ues always exist. This result is then used to show that each linear
operator on a complex vector space has an upper-triangular ma-
trix with respect to some basis. Similar techniques are used to
show that every linear operator on a real vector space has an in-
variant subspace of dimension 1 or 2. This result is used to prove
that every linear operator on an odd-dimensional real vector space
has an eigenvalue. All this is done without defining determinants
or characteristic polynomials!

• Inner-product spaces are defined in Chapter 6, and their basic
properties are developed along with standard tools such as ortho-
normal bases, the Gram-Schmidt procedure, and adjoints. This
chapter also shows how orthogonal projections can be used to
solve certain minimization problems.

• The spectral theorem, which characterizes the linear operators for
which there exists an orthonormal basis consisting of eigenvec-
tors, is the highlight of Chapter 7. The work in earlier chapters
pays off here with especially simple proofs. This chapter also
deals with positive operators, linear isometries, the polar decom-
position, and the singular-value decomposition.