Page 225 - A Course in Linear Algebra with Applications

7.2: Inner Product Spaces 209

14. Show that the vector equation of the plane through the
point (xo, yo, ZQ) with normal vector N is

T
{X - X 0) N = 0

where X and XQ are the vectors with entries x, y, z and xo,
y 0, z 0, respectively.
15. Prove the Cauchy-Schwartz Inequality for complex scalar
n
products in C .
16. Prove the Triangle Inequality for complex scalar products
n
i n C .
17. Establish the following expression for the vector triple
3
product in R : X x (Y x Z) = (X • Z)Y - (X • Y)Z. [Hint:
note that the vector on the right hand side is orthogonal to
to both X and Y x Z.\

7.2 Inner Product Spaces

We have seen how to introduce the notion of orthogonal-
n n
ity in the vector spaces R and C for arbitrary n. But what
about other vector spaces such as vector spaces of polynomials
or continuous functions? It turns out that there is a general
concept called an inner product which is a natural extension
n
of the scalar products in R n and C . This allows the intro-
duction of orthogonality in arbitrary real and complex vector
spaces.
Let V be a real vector space, that is, a vector space over
R. An inner product on V is a rule which assigns to each pair
of vectors u and v of V a real number < u, v >, their inner
product, such that the following properties hold:

(i) < v, v > > 0 and < v, v > = 0 if and only if v = 0;
(ii) < u, v > = < v, u >;
(iii)< cu + dv, w >= c < u, w > + <i < v, w > .