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Unit 27: Introduction and Forms on Inner Product Spaces
27.4 Keywords Notes
A Sesquilinear Form: A sesquilinear form on a real or complex vector space V is a function f on
V V with values in the field of scalars such that
f(c + , ) = cf(, ) + f(, )
f( + c, ) = cf(, ) + f(, )
for all , in V and all scalars c.
Hermitian: A form f on a real or complex vector space V is called Hermitian if
f(, ) = ( , )f
for all and in V.
Self-adjoint: The linear operator T is self-adjoint on a complex finite-dimensional inner product
space V, if and only if (T|) is real for every in V.
27.5 Review Questions
1. Let
1 i
A =
i 2
and let g be the form (on the space of 2 × 1 complex matrices) defined by g(X, Y) = Y*AX.
Is g an inner product?
2. Let f be the form on R defined by
2
f [(x , y ), (y , y )] = x y + x y
1 1 2 2 1 1 2 2
Find the matrix of f in each of the following bases:
{(1, –1), (1, 1)}, {(1, 2), (3, 4)}
Answer: Self Assessment
1. (b), (c)
27.6 Further Readings
Books Kenneth Hoffman and Ray Kunze, Linear Algebra
Michael Artin, Algebra
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