Page 225 - 35Linear Algebra

12

Eigenvalues and Eigenvectors

In a vector space with no structure other than the vector space rules, no
vector other than the zero vector is any more important than any other.
Once one also has a linear transformation the situation changes dramatically.
We begin with a fun example, of a type bound to reappear in your future
scientific studies:

String Theory Consider a vibrating string, whose displacement at point x at time t
is given by a function y(x, t):

The set of all displacement functions for the string can be modeled by a vector space
∂ k+m y(x, t)
2

V = y : R → R all partial derivatives exist .

∂x ∂t
k m
2
∂ y
The concavity and the acceleration of the string at the point (x, t) are 2 (x, t) and
∂x
2
∂ y
∂t 2 (x, t) respectively. Since quantities must exist at each point on the string for the
wave equation to make sense, we required that all partial derivatives of y(x, t) exist.
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