Direct sums of subspaces
The definition of an orthogonal sum requires two things: 1) every element
can be decomposed as
with
and 2) every element of
is orthogonal to every element of
Orthogonality of
to
implies
which, in turn, guarantees uniqueness of the representation
If we drop the orthogonality requirement but retain 1) and
we get the definition of a direct sum.
Definition. Let be two subspaces such that
The set
is called a direct sum of
and denoted
. The condition
provides uniqueness of the representation
Exercise 1. Let If
is decomposed as
with
define
Then
is linear and satisfies
.
Under conditions of Exercise 1, is an oblique projector of
onto
parallel to
Exercise 2. Prove that dimension additivity extends to direct sums: if then
Exercise 3. Let be two subspaces of
and suppose
Then to have
it is sufficient to check that
in which case
Proof. Denote By Exercise 2,
If
is not empty, then we can complete a basis in
with a nonzero vector from
to see that
which is impossible.
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