Quadratic Forms

Quadratic forms are mathematical objects that arise in a variety of areas in mathematics, including linear algebra, number theory, and geometry. In linear algebra, a quadratic form is a homogeneous polynomial of degree two in several variables.

Definition

Let $V$ be a vector space over a field $\mathbb{F}$. A quadratic form $Q:V\to \mathbb{F}$ is a map that associates to each vector $v\in V$ a scalar $Q(v)\in \mathbb{F}$, such that $Q(\lambda v)={\lambda }^{2}Q(v)$ for all $\lambda \in \mathbb{F}$ and $v\in V$, and the map $B:V\times V\to \mathbb{F}$ defined by $B(v,w)=Q(v+w)-Q(v)-Q(w)$ is a bilinear form.

Examples

1. In ${\mathbb{R}}^n$, the standard quadratic form is given by $Q(x_1,x_2,\dots ,x_n)={\sum }_{i=1}^n{a}_{ii}{x}_i^2+2{\sum }_{1\le i<j\le n}{a}_{ij}{x}_i{x}_j$, where $a_{ii}$ and $a_{ij}$ are constants.

2. In number theory, the quadratic form $Q(x,y)=a{x}^2+bxy+c{y}^2$ plays a crucial role in solving problems related to Diophantine equations.

Historical Context

Quadratic forms have a long history, dating back to ancient Greek mathematicians. The study of quadratic forms was greatly influenced by Indian mathematicians, particularly in the context of solving Diophantine equations. In the 18th and 19th centuries, mathematicians such as Gauss and Lagrange made significant contributions to the theory of quadratic forms, laying the foundations for their modern study.

Real-life Applications

Quadratic forms find applications in various real-life scenarios, such as in physics to describe the kinetic energy of a system, in economics to model utility functions, and in computer science for error-correcting codes.

Exam Questions

1. Consider the quadratic form $Q(x,y)=3x^2-2xy+4y^2$. Show that $Q$ is a positive definite form.

2. Explain how the discriminant of a quadratic form is related to its equivalence classes under proper equivalence.

3. Discuss the role of quadratic forms in cryptography and their applications in constructing cryptographic protocols.