Page 7 - Vector Analysis

§1.2 Inner Products and Inner Product Spaces 3

Example 1.9. The space Fn is n-dimensional, and C ([0, 1]) is infinitely dimensional (since
1, x, ¨ ¨ ¨ , xn´1 are n linearly independent vectors in C ([0, 1])).

1.1.3 Bases of a vector space

Definition 1.10 (Basis). Let V be a vector space over F. A set of vectors tviuiPI in V is
called a basis of V if for every v P V, there exists a unique tαiuiPI Ď F such that

v = ÿ αivi .

αPI

For a given basis B = tviuiPI, the coefficients tαiuiPI given in the above relation is denoted
by [v]B.

Example 1.11 (Standard Basis of Fn). Let ei = (0, , ¨ ¨ ¨ , 0, 1, 0, ¨ ¨ ¨ , 0), where 1 locates at
the i-th slot. Then the collection teiuin=1 is a basis of the vector space Fn over F since

n @ αi P F.

(α1, ¨ ¨ ¨ , αn) = ÿ αiei

i=1

The collection teiuni=1 is called the standard basis of Fn.

Example 1.12. Even though ␣1, x, ¨ ¨ ¨ , xk, ¨ ¨ ¨ ( is a set of linearly independent vectors, it
is not a basis of C ([0, 1]). However, let P([0, 1]) be the collection of polynomials defined on
[0, 1]. Then P([0, 1]) is still a vector space, and ␣1, x, ¨ ¨ ¨ , xk, ¨ ¨ ¨ ( is a basis of P([0, 1]).

1.2 Inner Products and Inner Product Spaces

Definition 1.13 (Inner product space). Let F = R or C. A vector space V over a scalar
field F with a bilinear form (¨, ¨) : V ˆ V Ñ F is called an inner product space if the
bilinear form satisfies

1. (v, v) ě 0 for all v P V.

2. (v, v) = 0 if and only if v = 0.

3. (v, w) = (w, v) for all v, w P V, where the bar over the scalar (w, v) is the complex
conjugate.

4. (v + w, u) = (v, u) + (w, u) for all u, v, w P V.