Page 227 - 35Linear Algebra

12.1 Invariant Directions 227

as well as a new linear function

d 2
L := : U −→ U .
dx 2

The number ω is called an angular frequency in many contexts, lets call its square λ :=
2
−ω to match notations we will use later (notice that for this particular problem λ must
2
2
2
then be negative). Then, because we want W(y) = 0, which implies d f/dx = ω f,
it follows that the vector v(x) ∈ U determining the vibrating string’s shape obeys

L(v) = λv .

This is perhaps one of the most important equations in all of linear algebra! It is
the eigenvalue-eigenvector equation. In this problem we have to solve it both for λ,
to determine which frequencies (or musical notes) our string likes to sing, and the
vector v determining the string’s shape. The vector v is called an eigenvector and λ
its corresponding eigenvalue. The solution sets for each λ are called V λ . For any λ the
set V k is a vector space since elements of this set are solutions to the homogeneous
equation (L − λ)v = 0.

We began this chapter by stating “In a vector space, with no other struc-
ture, no vector is more important than any other.” Our aim is to show you
that when a linear operator L acts on a vector space, vectors that solve the
equation L(v) = λv play a central role.

12.1 Invariant Directions

Have a look at the linear transformation L depicted below:

227