Summary: Lecture 15: Some remarks on dimensions of subspaces
Let U, W be subspaces of some finite dimensional vector space V . Then
dim(U + W) = dim(U) + dim(W) - dim(U W).
The proof is as follows: We start with the case where U W = {0}.
By problem (2) of assignment #10 both U and W are finite dimensional. Let
u1, . . . , un and w1, . . . , wm be bases of U and W respectively. We claim that
u1, . . . , un, w1, . . . , wm is a basis of U + W. Each vector in U + W is of the form
u + w, where u U and w W. On the other hand,
u =
n
i=1
iui and w =
m
j=1
jwj
for suitable scalars 1, . . . , n, 1, . . . , m. Then
u + w =
n
i=1
iui +

Source: Abbas, Casim - Department of Mathematics, Michigan State University

Collections: Mathematics