Page 342 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes The polarization Identity helps in determining the symmetric bilinear form by its associated
quadratic form.

30.5 Review Questions

1. Let V be a finite-dimensional vector space over a subfield F of the complex numbers, and
let S be the set of all symmetric bilinear forms on V.
(a) Show that S is a subspace of L(V, V, F)

(b) Find Dim S’
2
2. Let q be the quadratic form on R given by
q(x , x ) = 2bx x
1 2 1 2
2
Find an invertible linear operator V on R such that
2
2
+
(V q) (x , x ) = 2bx – 2bx .
1 2 1 2
Answers: Self Assessment
1. (b) and (c)
4. (a) f( , ) = ax y
1 1
(b) f( , ) = x y + 9x y
1 1 2 2
b
(c) f( , ) = (x y + y x )
2 1 2 1 2
Here = (x , x )
1 2
= (y , y )
1 2
30.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
Michael Artin, Algebra

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