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192 ALGEBRA

5.3.3-2. Relations between coordinate transformations and basis transformations.

Suppose that in a linear n-dimensional space V, the transition from its basis e 1 , ... , e n to
another basis 2 e 1 , ... , 2 e n is determined by the matrix A (see Paragraph 5.3.3-1). Let x be
any element of the space V with the coordinates (x 1 , ... , x n ) in the basis e 1 , ... , e n and the
coordinates (2x 1 , ... , 2x n ) in the basis 2 e 1 , ... , 2 e n , i.e.,

x = x 1 e 1 + ··· + x n e n = 2x 1 e 1 + ··· + 2x n e n .
2
2
Then using formulas (5.3.3.1), we obtain the following relations between these coordinates:

n n

x j = 2 x i a ij , 2 x k = x l b lk , j, k = 1, ... , n.
i=1 l=1

In terms of matrices and row vectors, these relations can be written as follows:
–1
(x 1 , ... , x n )= (2x 1 , ... , 2x n )A, (2x 1 , ... , 2x n )= (x 1 , ... , x n )A

or, in terms of column vectors,

T
T
T
T
–1 T
T
(x 1 , ... , x n ) = A (2x 1 , ... , 2x n ) , (2x 1 , ... , 2x n ) =(A ) (x 1 , ... , x n ) ,
where the superscript T indicates the transpose of a matrix.
5.4. Euclidean Spaces
5.4.1. Real Euclidean Space
5.4.1-1. Definition and properties of a real Euclidean space.

A real Euclidean space (or simply, Euclidean space) is a real linear space V endowed with a
scalar product (also called inner product), which is a real-valued function of two arguments
x V, y V called the scalar product of these elements, denoted by x ⋅ y, and satisfying the
following conditions (axioms of the scalar product):
1. Symmetry: x ⋅ y = y ⋅ x.
2. Distributivity: (x 1 + x 2 ) ⋅ y = x 1 ⋅ y + x 2 ⋅ y.
3. Homogeneity: (λx) ⋅ y = λ(x ⋅ y) for any real λ.
4. Positive definiteness: x ⋅ x ≥ 0 for any x,and x ⋅ x = 0 if and only if x = 0.
If the nature of the elements and the scalar product is concretized, one obtains a specific
Euclidean space.

Example 1. Consider the linear space B 3 of all free vectors in three-dimensional space. The space B 3
becomes a Euclidean space if the scalar product is introduced as in analytic geometry (see Paragraph 4.5.3-1):
x ⋅ y = |x||y| cos ϕ,
where ϕ is the angle between the vectors x and y.
n
Example 2. Consider the n-dimensional coordinate space R whose elements are ordered systems of n
arbitrary real numbers, x =(x 1, .. . , x n). Endowing this space with the scalar product
x ⋅ y = x 1y 1 + ·· · + x ny n,
we obtain a Euclidean space.