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190 ALGEBRA

By definition, the dimension of an affine space A is the dimension of the associated
linear space V,dim A =dim V.
Any linear space may be regarded as an affine space.
n
In particular, the space R can be naturally considered as an affine space. Thus if A =
n
(a 1 , ... , a n )and B =(b 1 , ... , b n ) are points of the affine space R , then the corresponding
−−→ n −−→
vector AB from the linear space R is defined by AB =(b 1 – a 1 , ... , b n – a n ).
Let A be an n-dimensional affine space with the associated linear space V.A coordinate
system in the affine space A is a fixed point O A, together with a fixed basis e 1 , ... ,
e n V. The point O is called the origin of this coordinate system.
Let M be a point of an affine space A with a coordinate system Oe 1 ... e n . One says
that the point M has affine coordinates (or simply coordinates) x 1 , ... , x n in this coordinate
system, and one writes M =(x 1 , ... , x n )if x 1 , ... x n are the coordinates of the radius-vector
−−→ −−→
OM in the basis e 1 , ... , e n , i.e., OM = x 1 e 1 + ··· + x n e n .

5.3.2. Subspaces of Linear Spaces
5.3.2-1. Concept of a linear subspace and a linear span.

A subset L of a linear space V is called a linear subspace of V if the following condi-
tions hold:
1. If x and y belong to L, then the sum x + y belongs to L.
2. If x belongs to L and λ is an arbitrary scalar, then the element λx belongs to L.
The null subspace in a linear space V is its subset consisting of the single element zero.
The space V itself can be regarded as its own subspace. These two subspaces are called
improper subspaces. All other subspaces are called proper subspaces.
Example 1. A subset B 2 consisting of all free vectors parallel to a given plane is a subspace in the linear
space B 3 of all free vectors.

The linear span L(x 1 , ... , x m )ofvectors x 1 , ... , x m in a linear space V is, by definition,
the set of all linear combinations of these vectors, i.e., the set of all vectors of the form
α 1 x 1 + ··· + α m x m ,
where α 1 , ... , α m are arbitrary scalars. The linear span L(x 1 , ... , x m ) is the least subspace
of V containing the elements x 1 , ... , x m .
If a subspace L of an n-dimensional space V does not coincide with V,then dim L <
n =dim V.
Let elements e 1 , ... , e k form a basis in a k-dimensional subspace of an n-dimensional
linear space V. Then this basis can be supplemented by elements e k+1 , ... , e n of the space
V, so that the system e 1 , ... , e k , e k+1 , ... , e n forms a basis in the space V.
THEOREM OF THE DIMENSION OF A LINEAR SPAN. The dimension of a linear span
L(x 1 , ... , x m ) of elements x m , ... , x m is equal to the maximal number of linearly indepen-
dent vectors in the system x 1 , ... , x m .

5.3.2-2. Sum and intersection of subspaces.
The intersection of subspaces L 1 and L 2 of one and the same linear space V is, by definition,
the set of all elements x of V that belong simultaneously to both spaces L 1 and L 2 .Such
elements form a subspace of V.
The sum of subspaces L 1 and L 2 of one and the same linear space V is, by definition,
the set of all elements of V that can be represented in the form y + z,where y is an element
of V 1 and z is an element of L 2 . The sum of subspaces is also a subspace of V.