Page 271 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

 With the help of a few examples the concept of inner product is illustrated. Notes
 The inner product is also related to the polarization identities.
 The relation between the vector space and the inner product is established.
 The Cauchy-Schwarz inequality is established.

 The Gram-Schmidt orthogonalization process help us to find a set of orthogonal vectors as
a bases of the vector space V.

24.4 Keywords

An Inner Product Space is a real or complex vector space, together with a specified inner product
on that space.
An Orthogonal Set: If S is a set of vectors in V, S is called an orthogonal set provided all pairs of
distinct vectors in S are orthogonal. An orthonormal set is an orthogonal set S with the additional
property that 1 for every d in S.
Bessel’s Inequality: Let ( , , . . . , ) be an orthogonal set of non-zero vectors in an inner
1 2 n
( | ) 2 2
product space V. If is any vector in V, then the Bessel Inequality is given by k 2 .
2
k k
Cauchy-Schwarz Inequality: If V is an inner product space, then for any vectors , in V,

( | ) ,

is called the Cauchy-Schwarz inequality and the above equality occurs if and only if and are

linearly dependent.

Conjugate Transpose Matrix: The conjugate transpose matrix B* is defined by the relation
B * B , where B is complex conjugate of the matrix B.
Kj jK
Gram-Schmidt Orthogonalization Process: Let V be an inner product space and let , , . . .
1 2 n
be any independent set of vectors in V, then one may construct orthogonal vectors , , . . .
1 2 n
in V by means of a construction known as Gram-Schmidt orthogonalization process.
Linearly Independent: An orthogonal set of non-zero vectors is linearly independent.
Polarization Identities: For the real vector space polarization identities are defined by

1 2 1 2
( | ) .
4 4
Standard Inner Product: If = (x , x , . . . x ), = (y , y , . . . y ) are vectors in F , there is an inner
n
1 2 n 1 2 n
product which we call the standard inner product, defined by the relation
n
( | ) x y i .
i
i 1
The Orthogonal Complement: 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.

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