Page 387 - Matrix Analysis & Applied Linear Algebra

5.9 Complementary Subspaces 383
5.9 COMPLEMENTARY SUBSPACES

The sum of two subspaces X and Y of a vector space V was defined on p. 166
to be the set X + Y = {x + y | x ∈X and y ∈Y}, and it was established that
X + Y is another subspace of V. For example, consider the two subspaces of
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shown in Figure 5.9.1 in which X is a plane through the origin, and Y is a
line through the origin.

Figure 5.9.1
Notice that X and Y are disjoint in the sense that X∩ Y = 0. The paral-
3
lelogram law for vector addition makes it clear that X + Y = because each
3
vector in can be written as “something from X plus something from Y. ”
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Thus is resolved into a pair of disjoint components X and Y. These ideas
generalize as described below.

Complementary Subspaces
Subspaces X, Y of a space V are said to be complementary whenever

V = X + Y and X∩ Y = 0, (5.9.1)
in which case V is said to be the direct sum of X and Y, and this is
denoted by writing V = X⊕ Y.
• Foravector space V with subspaces X, Y having respective bases
B X and B Y , the following statements are equivalent.
V = X⊕ Y. (5.9.2)
For each v ∈V there are unique vectors x ∈X and y ∈Y
such that v = x + y. (5.9.3)
B X ∩B Y = φ and B X ∪B Y is a basis for V. (5.9.4)