Constructing a projector onto a given subspace
Let be a subspace of
Let
and fix some basis
in
Define the matrix
of size
(the vectors are written as column vectors).
Exercise 1. a) With the above notation, the matrix exists. b) The matrix
exists. c)
is a projector.
Proof. a) The determinant of is not zero by linear independence of the basis vectors, so its inverse
exists. We also know that
and its inverse are symmetric:
(1)
b) To see that exists just count the dimensions.
c) Let's prove that is a projector. (1) allows us to make the proof compact.
is idempotent:
is symmetric:
Exercise 2. projects onto
:
Proof. First we show that Put
(2)
for any Then
(3)
This shows that
Let's prove the opposite inclusion. Any element of is of form
(4)
with some Plugging (4) in (3) we have
which shows that
Exercise 3. Let be a subspace of
and let
. Then
.
Proof. Since any element of is orthogonal to any element of
, we have only to show that any
can be represented as
with
. Let
be the projector from Exercise 1, where
. Put
,
.
is obvious. For any
,
, so
is orthogonal to
. By definition of
, we have
.
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