Page 226 - 35Linear Algebra

226 Eigenvalues and Eigenvectors

Note also that the function y(x, t) = 0 —drawn in grey—is the only special vector in
the vector space V .
We now add some extra information. The string’s behavior in time and space can
be modeled by a wave equation

2
2
∂ y ∂ y
= ,
∂t 2 ∂x 2
which says that the acceleration of a point on the string is equal its concavity at that
point. For example, if the string were made of stretched rubber, it would prefer to be
in a straight line, so this equation makes good intuitive sense. Not all of the functions
in V are solutions to the wave equation; not all of the functions in the vector space V
describe the way a string would really vibrate. The ways a string would really vibrate
are (at least approximately) solutions to the wave equation above, which can rewritten
as a linear function
Wy = 0
where
2 2
∂ ∂
W = − + : V → V .
∂t 2 ∂x 2
Some examples of solutions are

y 1 (x, t) = sin(t) sin(x) y 2 (x, t) = 3 sin(2t) sin(2x)

and
y 3 (x, t) = sin(t) sin(x) + 3 sin(2t) sin(2x) .

Since Wy = 0 is a homogeneous linear equation, linear combinations of solutions are
solutions; in other words the kernel ker(w) is a vector space. Given the linear function
W, some vectors are now more special than others.
We can use musical intuition to do more! If the ends of the string were held
fixed, we suspect that it would prefer to vibrate at certain frequencies corresponding
to musical notes. This is modeled by looking at solutions of the form

y(x, t) = sin(ωt)v(x) .

Here the periodic sine function accounts for the string’s vibratory motion, while the
function v(x) gives the shape of the string at any fixed instant of time. Observe that

2
d f
2
W sin(ωt)v(x) = sin(ωt) + ω f .
dx 2
This suggests we introduce a new vector space
k
d f

U = v : R → R all derivatives exist ,

dx k
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