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5.3. LINEAR SPACES 191

THEOREM. The sum of dimensions of arbitrary subspaces L 1 and L 2 of a finite-
dimensional space V is equal to the sum of the dimension of their intersection and the
dimension of their sum.
Example 2. Let V be the linear space of all free vectors (in three-dimensional space). Denote by L 1 the
subspace of all free vectors parallel to the plane OXY , and by L 2 the subspace of all free vectors parallel to
the plane OXZ. Then the sum of the subspaces L 1 and L 2 coincides with V, and their intersection consists of
all free vectors parallel to the axis OX.
The dimension of each space L 1 and L 2 is equal to two, the dimension of their sum is equal to three, and
the dimension of their intersection is equal to unity.

5.3.2-3. Representation of a linear space as a direct sum of its subspaces.

A linear space V can be represented as a direct sum of its subspaces, V 1 and V 2 if each
element x V admits the unique representation x = x 1 + x 2 ,where x 1 V 1 and x 2 V 2 .In
this case, one writes V = V 1 ⊕ V 2 .
Example 3. The space V of all free vectors (in three-dimensional space) can be represented as the direct
sum of the subspace V 1 formed by all free vectors parallel to the plane OXY and the subspace V 2 formed by
all free vectors parallel to the axis OZ.
THEOREM. An n-dimensional space V is a direct sum of its subspaces V 1 and V 2 if and
only if the intersection of V 1 and V 2 is the null subspace and dim V =dim V 1 +dim V 2 .
Remark. If R is the sum of its subspaces R 1 and R 2, but not the direct sum, then the representation
x = x 1 + x 2 is nonunique, in general.

5.3.3. Coordinate Transformations Corresponding to Basis
Transformations in a Linear Space

5.3.3-1. Basis transformation and its inverse.
Let e 1 , ... , e n and 2 e 1 , ... , 2 e n be two arbitrary bases of an n-dimensional linear space V.
Suppose that the elements 2 e 1 , ... , 2 e n are expressed via e 1 , ... , e n by the formulas
2 e 1 = a 11 e 1 + a 12 e 2 + ··· + a 1n e n ,
2 e 2 = a 21 e 1 + a 22 e 2 + ··· + a 2n e n ,
.. ........ ........ ........ ......
2 e n = a n1 e 1 + a n2 e 2 + ··· + a nn e n .

Thus, the transition from the basis e 1 , ... , e n to the basis 2 e 1 , ... , 2 e n is determined by the
matrix
⎛ ⎞
a 11 a 12 ··· a 1n
⎜ a 21 a 22 ··· a 2n ⎟
A ≡ ⎝ . . . . . . . . . . . . ⎠ .
⎜
⎟
a n1 a n2 ··· a nn
Note that det A ≠ 0, i.e., the matrix A is nondegenerate.
The transition from the basis 2 e 1 , ... , 2 e n to the basis e 1 , ... , e n is determined by the
–1
matrix B ≡ [b ij ]= A . Thus, we can write
n n

2 e i = a ij e j , e k = b kj e j (i, k = 1, 2, ... , n). (5.3.3.1)
2
j=1 j=1