Page 149 - A Course in Linear Algebra with Applications

5.3: Operations with Subspaces 133

5.3 Operations with Subspaces
If U and W are subspaces of a vector space V, there
are two natural ways of combining U and W to form new
subspaces of V. The first of these subspaces is the intersection

unw,

which is the set of all vectors that belong to both U and V.
The second subspace that can be formed from U and W
is not, as one might perhaps expect, their union U UW; for
this is not in general closed under addition, so it may not be
a subspace. The subspace we are looking for is the sum

U + W,

which is denned to be the set of all vectors of the form u + w
where u belongs to U and w to W.
The first point to note is that these are indeed subspaces.

Theorem 5.3.1
If U and W are subspaces of a vector space V, then U C\W
and U + W are subspaces of V.
Proof
Certainly U D W contains the zero vector and it is closed with
respect to addition and scalar multiplication since both U and
W are; therefore U fl W is a subspace.
The same method applies to U + W. Clearly this contains
0 + 0 = 0. Also, if Ui, U2 and wi, w 2 are vectors in U and
W respectively, and c is a scalar, then

(ui + wi) + (u 2 + w 2 ) = (ui + u 2 ) + (wi + w 2 )

and
c(ui + w i ) = cui + cwi,