Page 150 - A Course in Linear Algebra with Applications

134 Chapter Five: Basis and Dimension

both of which belong to U + W. Thus U + W is closed with
respect to addition and scalar multiplication and so it is a
subspace.
Example 5.3.1

Consider the subspaces U and W of R 4 consisting of all vectors
of the forms
a
( \ f°\
b d
and
c e
Vo/ \fJ
respectively, where a, b, c, d, e are arbitrary scalars. Then
U n W consists of all vectors of the form

ft)

c '
w

while U + W equals R 4 since every vector in R 4 can be ex-
pressed as the sum of a vector in U and a vector in W.

For subspaces of a finitely generated vector space there is
an important formula connecting the dimensions of their sum
and intersection.

Theorem 5.3.2
Let U and W be subspaces of a finitely generated vector space
V. Then

dim(U + W)+ dim(U n W) = dim(U) + dim(W).

Proof
If U = 0, then obviously U + W = W and U n W = 0; in this
case the formula is certainly true, as it is when W = 0.