Page 6 - Vector Analysis

2 CHAPTER 1. Linear Algebra

Example 1.3. Let F be a scalar field. The space Fn is the collection of n-tuple v =
(v1, v2, ¨ ¨ ¨ , vn) with vi P F with addition + and scalar multiplication ¨ defined by

(v1, ¨ ¨ ¨ , vn) + (w1, ¨ ¨ ¨ , wn) ” (v1 + w1, ¨ ¨ ¨ , vn + wn) ,
α(v1, ¨ ¨ ¨ , vn) ” (αv1, ¨ ¨ ¨ , αvn) .

Then Fn is a vector space.

Example 1.4. Let F = R or C, and V be the collection of all R-valued continuous functions
on [0, 1]. The vector addition + and scalar multiplication ¨ is defined by

(f + g)(x) = f (x) + g(x) @ f, g P V ,
(α ¨ f )(x) = αf (x) @f P V,α P F.

Then V is a vector space, and is denoted by C ([0, 1]; F). When the scalar field under
consideration is clear, we simply use C ([0, 1]) to denote this vector space.

Definition 1.5 (Vector subspace). Let V be a vector space over scalar field F. A subset
W Ď V is called a vector subspace of V if itself is a vector space over F.

1.1.1 The linear independence of vectors

Definition 1.6. Let V be a vector space over a scalar field F. k vectors v1, v2, ¨ ¨ ¨ , vk in V
is said to be linearly dependent if there exists (α1, ¨ ¨ ¨ , αk) Ď Fk, (α1, ¨ ¨ ¨ , αk) ‰ 0 such
that α1v1 + α2v2 + ¨ ¨ ¨ + αkvk = 0. k vectors v1, v2, ¨ ¨ ¨ , vk in V is said to be linearly
independent if they are not linearly dependent. In other words, tv1, ¨ ¨ ¨ , vku are linearly
independent if

α1v1 + α2v2 + ¨ ¨ ¨ + αkvk = 0 ñ α1 = α2 = ¨ ¨ ¨ = αk = 0 .

Example 1.7. The k vectors t1, x, x2, ¨ ¨ ¨ , xk´1u are linearly independent in C ([0, 1]) for
all k P N.

1.1.2 The dimension of a vector space

Definition 1.8. The dimension of a vector space V is the number of maximum linearly
independent set in V, and in such case V is called an n-dimensional vector space, where
n the the dimension of V. If for every number n P N there exists n linearly independent
vectors in V, the vector space V is said to be infinitely dimensional.