Page 269 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

also belongs to W . Since cE + E is a vector in W, it follows from Theorem 4 that Notes
E(c + ) = cE +E .
Of course, one may also prove the linearity of E by using (11) . Again let be any vector in V.
Then E is the unique vector in W such that – E is in W . Thus E = 0 when is in W .
Conversely, is in W when E = 0. Thus W is the null space of E. The equation
= E + – E

show that V = W + W ; moreover, M W = {0}. For if is vector in M W , then ( | ) = 0.
Therefore, = 0, and V is the direct sum of W and W .
Corollary: Under the conditions of the theorem, I – E is orthogonal projection of V on W . It is
an idempotent linear transformation of V onto W with null space W.
Proof: We have already seen that the mapping E is the orthogonal projection of V on
W . Since E is a linear transformation, this projection on W is the linear transformation I – E.
From its geometric properties one sees that I – E is an idempotent transformation of V onto W.
This also follows from the computation
(I – E) (I – E) = I – E – E + E 2

= I – E.
Moreover, (I – E) = 0 if and only if = E , and this is the case if and only if is in W. Therefore
W is the null space of I – E.

The Gram-Schmidt process may now be described geometrically in the following way. Given an
inner product space V and vectors , . . . , in V, let P (k > 1) be the orthogonal projection of V
1 n k
on the orthogonal complement of the subspace spanned by , . . . , , and set P = I. Then the
1 k – 1 1
vectors one obtains by applying the orthogonalization process to , . . . , , are defined by the
1 n
equations
= P , 1 k n .
k k k
Theorem 5 implies another result known as Bessel’s inequality.
Corollary: Let { , . . ., } be an orthogonal set of non-zero vectors in an inner product space V.
1 n
If is any vector in V, then
2
( | )
k 2
2
k k
and equality holds if and only if
( | )
= k 2 k .
k k
2
Proof: Let = | / . Then = + where ( / ) = 0. Hence
k k k
k
2 2 2
= .
It now suffices to prove that

2
( | )
2 k
= 2 .
k k

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