Page 153 - A Course in Linear Algebra with Applications

5.3: Operations with Subspaces 137

10
Of course dim(R ) = 10 and, since U + W is a subspace of
10
R , its dimension cannot exceed 10. Therefore by 5.3.2

dim(C/ fl W) = dim(C7) + dim(W) - dim(U + W)
> 6 + 8 - 1 0 = 4.

So the dimension of the intersection is at least 4. The reader
is challenged to think of an example which shows that the
intersection really can have dimension 4.

Direct sums of subspaces
Let U and W be two subspaces of a vector space V. Then
V is said to be the direct sum of U and W if

V = U + W and UnW = 0.

The notation for the direct sum is

v = u®w.

Notice the consequence of the definition: each vector v of V
has a unique expression of the form v = u + w where u belongs
to U and w to W. Indeed, if there are two such expressions
v = ui + wi = U2 + W2 with Uj in U and Wj in W, then
ui — u 2 = w 2 — wi, which belongs to U D W = 0; hence
ui = u 2 and wi = W2.

Example 5.3.3
Let U denote the subset of R 3 consisting of all vectors of the
form

(i)