Page 154 - A Course in Linear Algebra with Applications

138 Chapter Five: Basis and Dimension

and let W be the subset of all vectors of the form

( ! )
where a, b, c are arbitrary scalars. Then U and W are sub-
3
spaces of R . In addition U + W = R 3 and UC\W = 0. Hence
R 3 = u 8 W.

Theorem 5.3.3
If V is a finitely generated vector space and U and W are
subspaces of V such that V = U ® W, then

dim(V) = dim(C7) + dim(W).

This follows at once from 5.3.2 since dim(U DW) = 0 .

Direct sums of more than two subspaces
The concept of a direct sum can be extended to any finite
set of subspaces. Let U\, U2, • • •, £4 be subspaces of a vector
space V. First of all define the sum of these subspaces

t/i + • • • + U k

to be the set of all vectors of the form Ui + • • • + u& where
Uj belongs to Ui. This is clearly a subspace of V. The vector
space V is said to be the direct sum of the subspaces U\,... Uk,
in symbols
v = u @u ®---®u ,
1
k
2
if the following hold:
(i)V = U 1 + --- + U k;
(ii) for each i = 1, ,..., k the intersection of Ui with the
2
sum of all the other subspaces Uj, j ^ i, equals zero.