Page 29 - Linear Algebra Done Right

Sums and Direct Sums
must contain all finite sums of their elements). Thus U 1 + ··· + U m is
the smallest subspace of V containing U 1 ,...,U m .
Suppose U 1 ,...,U m are subspaces of V such that V = U 1 +···+U m . 15
Thus every element of V can be written in the form
u 1 +· · ·+ u m ,
where each u j ∈ U j . We will be especially interested in cases where
each vector in V can be uniquely represented in the form above. This
situation is so important that we give it a special name: direct sum.
Specifically, we say that V is the direct sum of subspaces U 1 ,...,U m ,
written V = U 1 ⊕···⊕U m , if each element of V can be written uniquely The symbol ⊕,
as a sum u 1 +· · ·+ u m , where each u j ∈ U j . consisting of a plus
Let’s look at some examples of direct sums. Suppose U is the sub- sign inside a circle, is
3
space of F consisting of those vectors whose last coordinate equals 0, used to denote direct
3
and W is the subspace of F consisting of those vectors whose first two sums as a reminder
coordinates equal 0: that we are dealing with
a special type of sum of
3 3
U ={(x, y, 0) ∈ F : x, y ∈ F} and W ={(0, 0,z) ∈ F : z ∈ F}. subspaces—each
element in the direct
3
Then F = U ⊕ W, as you should verify. sum can be represented
As another example, suppose U j is the subspace of F n consisting only one way as a sum
of those vectors whose coordinates are all 0, except possibly in the j th of elements from the
n
slot (for example, U 2 ={(0,x, 0,..., 0) ∈ F : x ∈ F}). Then
specified subspaces.
n
F = U 1 ⊕· · · U n ,
as you should verify.
As a final example, consider the vector space P(F) of all polynomials
with coefficients in F. Let U e denote the subspace of P(F) consisting
of all polynomials p of the form

2
p(z) = a 0 + a 2 z +· · ·+ a 2m z 2m ,
and let U o denote the subspace of P(F) consisting of all polynomials p
of the form

3
p(z) = a 1 z + a 3 z +· · ·+ a 2m+1 z 2m+1 ;
here m is a nonnegative integer and a 0 ,...,a 2m+1 ∈ F (the notations
U e and U o should remind you of even and odd powers of z). You should
verify that