Page 228 - A Course in Linear Algebra with Applications

212 Chapter Seven: Orthogonality in Vector Spaces

It follows from the definition of an inner product that the zero
vector is orthogonal to every vector and no non-zero vector can
be orthogonal to itself.

Example 7.2.4
Show that the functions sin x, m = 1,2,..., are mutually
orthogonal in the inner product space C[0, n] where the inner
product is given by the formula < f,g > = JQ f(x)g(x)dx.
We have merely to compute the inner product of sin mx
and sin nx :
r

< sin mx, sin nx > = sin mx sin nx dx.
Jo
Now, according to a well-known trigonometric identity,

sinmx sin nx = -(cos(m — n)x — cos(m + n)x).

Therefore, on evaluating the integrals, we obtain as the value
of < sin mx, sin nx >

[ — r sin(m — n)x — — sin(m -f n)x)7. — 0,
L v ; v ; J0
2(m-n) 2(m + n)
provided m ^ n. This is a very important set of orthogonal
functions which plays a basic role in the theory of Fourier
series.
If v is a vector in an inner product space V, then
< v,v > > 0, so this number has a real square root. This
allows us to define the norm of v to be the real number

||v|| = V< v,v>.

Thus ||v|| > 0 and ||v|| equals zero if and only if v = 0. A
vector with norm 1 is called a unit vector. It is clear that
norm is a generalization of length in Euclidean space.