Page 155 - A Course in Linear Algebra with Applications

5.3: Operations with Subspaces 139

In fact these are equivalent to requiring that every ele-
ment of V be expressible in a unique fashion as a sum of the
form ui + • • • + Ufc where u^ belongs to Ui.
The concept of a direct sum is a useful one since it often
allows us to express a vector space as a direct sum of subspaces
that are in some sense simpler.

Example 5.3.4

Let Ui,U2, U3 be the subspaces of R 5 which consist of all
vectors of the forms

/ d \
0 b 0
o > 0 > 0
0 c 0

respectively, where a, b, c, d, e are arbitrary scalars. Then
R 5 = Ui@U 2®U 3.

Bases for the sum and intersection of subspaces

Suppose that V is a vector space over a field F with posi-
tive dimension n and let there be given a specific ordered basis.
Assume that we have vectors u i , . . . , u r and w i , . . . , w s , gen-
erating subspaces U and W respectively. How can we find
bases for the subspaces U + W and UTiW and hence compute
their dimensions?
The first step in the solution is to translate the problem
n
to the vector space F . Associate with each Ui and Wj its
coordinate column vector Xi and Yj with respect to the given
ordered basis of V. Then X\,..., X r and Y\,..., Y s generate
n
respective subspaces U* and W* of F . It is sufficient if we
can find bases for U* + W* and U* D W* since from these
bases for U + W and U C\W can be read off. So assume from
n
now on that V equals F .