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# Basic algebraic concepts

Definition 2.1   Let be a vector space over a field . A function

is said to be a bilinear form if

for any vectors and scalar .

Definition 2.2   A bilinear form is said to be symmetric (resp. skew-symmetric) if (resp. ) for any vectors . The kernel of a bilinear form is the set of vectors for which for all . The dimension of the kernel is called the nullity of . If the nullity of a bilinear form is zero, the form is said to be nondegenerate.

Exercise 2.1 (10)   Let be a vector space over , let be its dual, and let denote the product . Show that the function

defined by is a skew-symmetric, nondegenerate bilinear form. This is the canonical symplectic form on .

Exercise 2.2 (*05)   Let be a vector space of dimension over a field with elements. How many different bilinear forms are there on ?

Exercise 2.3 (10)   Let be an -dimensional vector space over a field . Show that if we fix a basis of , bilinear forms are in one-to-one correspondence with matrices with entries in . Moreover, the form is symmetric if and only if the matrix is symmetric.

Definition 2.3   Let be a vector space over a field . A quadratic form on is a function such that there exists a bilinear form with for all

Exercise 2.4 (15)   Show that if is a quadratic form, there exists a unique symmetric bilinear form such that for all . Give a formula for in terms of .

Definition 2.4   A quadratic form on is said to be positive definite (resp. negative definite) if (resp. ) for all nonzero vectors .

Remark. If you did the previous exercises, you know that a quadratic form on is of the form

with .

Let be a vector space and let be its projectivization. If is a quadratic form on and is a nonzero vector such that , then for any scalar . This suggests the following definition:

Definition 2.5   Let be a vector space and let be its projectivization. If is a quadratic form on , the projective quadric associated to is the set

A projective quadric on is called a conic. If the symmetric bilinear form associated to is nondegenerate, we shall say that is nondegenerate or proper.

It may happen that the projective quadric associated to a quadratic form be empty. For example if , then is the empty set.

We end the section by proving the following important result:

Theorem 2.1   If is a nondegenerate quadric, the set of all hyperplanes tangent to is a nondegenerate quadric in the dual space .

Definition 2.6   The projective quadric in formed by the set of hyperplanes tangent to a nondegenderate quadric is called the dual quadric of and will be denoted by .

Let be the nondegenerate quadratic form associated to and let be the symmetric bilinear form satisfying . Define the map

by the equation .

Exercise 2.5 (00)   Show that is an invertible linear map.

Exercise 2.6 (10)   Let be a nonzero vector lying on the cone . Show that

is a hyperplane in that passes through the origin and is tangent to the cone at the point . Show that if is a nonzero scalar, .

Exercise 2.7 (05)   Prove that the dual quadric is the image of under the projective transformation induced by . Use this to prove theorem 2.1.

Exercise 2.8 (10)   Study the duality of degenerate quadrics.

Next: Action of on the Up: The Projective Geometry of Previous: Motivation
Juan Carlos Alvarez 2001-01-30