Page 267 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

for every in W. In particular, if is in W and , we may take Notes

( | )
= 2 ( ),

Then the inequality reduces to the statement
2 2
( | ) ( | )
2 2 2 0.

This holds if and only if ( – | – ) = 0. Therefore, – is orthogonal to every vector in W. This
completes the proof of the equivalence of the two conditions on a given in (i). This orthogonality
condition is evidently satisfied by at most one vector in W, which proves (ii).

Now suppose that W is a finite-dimensional subspace of V. Then we know, as a corollary of
Theorem 3, that W has an orthogonal basis. Let { , . . . , } be any orthogonal basis for W and
1 n
define by (11). Then, by the computation in the proof of Theorem 3, – is orthogonal to each
of the vectors ( – is vector obtained at the last stage when the orthogonalization process is
k
applied to , . . . , , ). Thus – is orthogonal to every linear combination of , . . . , , i.e,
1 n 1 n
to every vector in W. If is in W and , it follows that . Therfore, is the best
approximation to that lies in W.
Definition: Let V be an inner product space and S any set of vectors in V. The orthogonal
complement of S is the set S of all vectors in V which are orthogonal to every vector in S.
The orthogonal complement of V is the zero subspace, and conversely {0} V .If S is any subset
of V, its orthogonal complement S (S perp) is always a subspace of V. For S is non-empty, since
it contains 0; and whenever and are in S and c is any scalar,
(c | ) = ( | ) ( | )
c
= c0 + 0
= 0
for every in S, thus c + also lies in S. In Theorem 4 the characteristic property of the vector
is that it is the only vector in W such that – belongs to W .
Definition: Whenever the vector in Theorem 4 exists it is called the orthogonal projection of
on W. If every vector in V has an orthogonal projection on W, the mapping that assigns to each
vector in V its orthogonal projection on W is called the orthogonal projection of V on W.
By Theorem 4, the orthogonal projection of an inner product space on a finite-dimensional
subspace always exists. But Theorem 4 also implies the following result.
Corollary: Let V be an inner product space, W a finite-dimensional subspace, and E the orthogonal
projection of V on W. Then the mapping
E

is the orthogonal projection of V on W .

Proof: Let be an arbitrary vector in V. Then – E is in W , and for any in W , – = E +
( – E – ) . Since E is in W and – E – is in W , it follows that

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