Page 221 -

188 ALGEBRA

Example 1. Consider the set of all free vectors in three-dimensional space. If addition of these vectors
and their multiplication by scalars are defined as in analytic geometry (see Paragraph 4.5.1-1), this set becomes
a linear space denoted by B 3.
Example 2. Consider the set {x} whose elements are all positive real numbers. Let us define the sum of
two elements x and y as the product of x and y, and define the product of a real scalar λ and an element x as the
λth power of the positive real x. The number 1 is taken as the zero element of the space {x}, and the opposite
of x is taken equal to 1/x. It is easy to see that the set {x} with these operations of addition and multiplication
by scalars is a linear space.
n
Example 3. Consider the n-dimensional coordinate space R , whose elements are ordered sets of n
arbitrary real numbers (x 1, ... , x n). The generic element of this space is denoted by x, i.e., x =(x 1, ... , x n),
and the reals x 1, ... , x n are called the coordinates of the element x. From the algebraic standpoint, the set R n
may be regarded as the set of all row vectors with n real components.
n
The operations of addition of element of R and their multiplication by scalars are defined by the following
rules:
(x 1, ... , x n)+(y 1, .. . , y n)=(x 1 + y 1, .. . , x n + y n),
λ(x 1, ... , x n)=(λx 1, ... , λx n).
Remark. If the field of scalars λ, μ, ... in the above definition is the field of all real numbers, the
corresponding linear spaces are called real linear spaces.If the field of scalars is that of all complex numbers,
the corresponding space is called a complex linear space. In many situations, it is clear from the context which
field of scalars is meant.
The above axioms imply the following properties of an arbitrary linear space:
1. The zero vector is unique, and for any element x the opposite element is unique.
2. The zero vector 0 is equal to the product of any element x by the scalar 0.
3. For any element x, the opposite element is equal to the product of x by the scalar –1.
4. The difference of two elements x and y, i.e., the element z such that z + y = x, is unique.

5.3.1-2. Basis and dimension of a linear space. Isomorphisms of linear spaces.
An element y is called a linear combination of elements x 1 , ... , x k of a linear space V if
there exist scalars α 1 , ... , α k such that

y = α 1 x 1 + ··· + α k x k .

Elements x 1 , ... , x k of the space V are said to be linearly dependent if there exist scalars
2
2
α 1 , ... , α k such that |α 1 | + ··· + |α k | ≠ 0 and
α 1 x 1 + ··· + α k x k = 0,

where 0 is the zero element of V.
Elements x 1 , ... , x k of the space V are said to be linearly independent if for any scalars
2
2
α 1 , ... , α k such that |α 1 | + ··· + |α k | ≠ 0,we have
α 1 x 1 + ··· + α k x k ≠ 0.

THEOREM. Elements x 1 , ... , x k of a linear space V are linearly dependent if and only
if one of them is a linear combination of the others.
Remark. If at least one of the elements x 1, ... , x k is equal to zero, then these elements are linearly depen-
dent. If some of the elements x 1, ... , x k are linearly dependent, then all these elements are linearly dependent.
n
Example 4. The elements i 1 =(1, 0, ... , 0), i 2 =(0, 1, ... , 0), ... , i n =(0, 0, .. . , 1)ofthe space R (see
n
Example 3) are linearly independent. For any x =(x 1, ... , x n) R , the vectors x, i 1, .. . , i n are linearly
dependent.