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5.3. LINEAR SPACES 189

A basis of a linear space V is defined as any system of linearly independent vectors
e 1 , ... , e n such that for any element x of the space V there exist scalars x 1 , ... , x n such
that
x = x 1 e 1 + ··· + x n e n .

This relation is called the representation of an element x in terms of the basis e 1 , ... , e n ,
and the scalars x 1 , ... , x n are called the coordinates of the element x in that basis.

UNIQUENESS THEOREM. The representation of any element x V in terms of a given
basis e 1 , ... , e n is unique.

Let e 1 , ... , e n be any basis in V and vectors x and y have the coordinates x 1 , ... , x n and
y 1 , ... , y n in that basis. Then the coordinates of the vector x + y in that basis are x 1 + y 1 ,
... , x n + y n , and the coordinates of the vector λx are λx 1 , ... , λx n for any scalar λ.

Example 5. Any three noncoplanar vectors form a basis in the linear space B 3 of all free vectors. The n
n
elements i 1 =(1, 0, ... , 0), i 2 =(0, 1, ... , 0), .. . , i n =(0, 0, ... , 1) form a basis in the linear space R .Any
basis of the linear space {x} from Example 2 consists of a single element. This element can be arbitrarily
chosen of nonzero elements of this space.
A linear spaceV issaidtoben-dimensional if it contains n linearly independent elements
and any n + 1 elements are linearly dependent. The number n is called the dimension of
that space, n =dim V.
A linear space V is said to be infinite-dimensional (dim V = ∞) if for any positive
integer N it contains N linearly independent elements.

THEOREM 1. If V is a linear space of dimension n,then any n linearly independent
elements of that space form its basis.

THEOREM 2. If a linear space V has a basis consisting of n elements, then dim V = n.
n
Example 6. The dimension of the space B 3 of all vectors is equal to 3. The dimension of the space R is
equal to n. The dimension of the space {x} is equal to 1.
Two linear spaces V and V over the same field of scalars are said to be isomorphic

if there is a one-to-one correspondence between the elements of these spaces such that if

elements x and y from V correspond to elements x and y from V , then the element x + y
corresponds to x + y and the element λx corresponds to λx for any scalar λ.

Remark. If linear spaces V and V are isomorphic, then the zero element of one space corresponds to the
zero element of the other.
THEOREM. Any two n-dimensional real (or complex) spaces V and V are isomorphic.

5.3.1-3. Affine space.
An affine space is a nonempty set A that consists of elements of any nature, called points,
for which the following conditions hold:
I. There is a given linear (vector) space V, called the associated linear space.
II. There is a rule by which any ordered pair of points A, B A is associated with an
−−→
element (vector) from V; this vector is denoted by AB and is called the vector issuing
from the point A with endpoint at B.
III. The following conditions (called axioms of affine space) hold:
1. For any point A A and any vector a V, there is a unique point B A such that
−−→
AB = a.
−−→ −−→ −→
2. AB + BC = AC for any three points A, B, C A.