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194 ALGEBRA

One says that n elements i 1 , ... , i n of an n-dimensional Euclidean space V form its
orthonormal basis if these elements have unit norm and are mutually orthogonal, i.e.,

1 for i = j,

i i ⋅ i j =
0 for i ≠ j.
THEOREM. In any n-dimensional Euclidean space V, there exists an orthonormal basis.

Orthogonalization of linearly independent elements:
Let e 1 , ... , e n be n linearly independent vectors of an n-dimensional Euclidean space V.
From these vectors, one can construct an orthonormal basis of V using the following
algorithm (called Gram–Schmidt orthogonalization):

i
g i
i i = √ , where g i = e i – (e i ⋅ i j )i j (i = 1, 2, ... , n). (5.4.1.3)
g i ⋅ g i
j=1
Remark. In any n-dimensional (n > 1) Euclidean space V,there existinfinitely many orthonormal bases.
Properties of an orthonormal basis of a Euclidean space:
1. Let i 1 , ... , i n be an orthonormal basis of a Euclidean space V. Then the scalar product
of two elements x = x 1 i 1 + ··· + x n i n and y = y 1 i 1 + ··· + y n i n is equal to the sum of
products of their respective coordinates:

x ⋅ y = x 1 y 1 + ··· + x n y n .

2. The coordinates of any vector x in an orthonormal basis i 1 , ... , i n are equal to the scalar
product of x and the corresponding vector of the basis (or the projection of the element
x on the axis in the direction of the corresponding vector of the basis):

x k = x ⋅ i k (k = 1, 2, ... , n).

Remark. In an arbitrary basis e 1, ... , e n of a Euclidean space, the scalar product of two elements
x = x 1e 1 + ··· + x ne n and y = y 1e 1 + ·· · + y ne n has the form
n n

x ⋅ y = a ijx iy j,
i=1 j=1
where a ij = e i ⋅ e j (i, j = 1, 2, ... , n).
Let X, Y be subspaces of a Euclidean space V. The subspace X is called the orthogonal
complement of the subspace Y in V if any element x of X is orthogonal to any element y of Y
and X ⊕ Y = V.

THEOREM. Any n-dimensional Euclidean space V can be represented as the direct sum
of its arbitrary subspace Y and its orthogonal complement X.

Two Euclidean spaces V and V are said to be isomorphic if one can establish a one-to-one
2
correspondence between the elements of these spaces satisfying the following conditions:
if elements x and y of V correspond to elements 2 x and 2 y of V, then the element x + y
2
corresponds to 2 x +2 y; the element λx corresponds to λ2 x for any λ; the scalar product (x⋅ y) V
is equal to the scalar product (2 x ⋅ 2 y) .
2 V
THEOREM. Any two n-dimensional Euclidean spaces V and V are isomorphic.
2