The Algebra of Linear Transfo...

CHAPTER I

The Algebra of Linear Transformations

and Quadratic Forms

In the present volume we shall be concerned with many topics in mathematical analysis which are intimately related to the theory of linear transformations and quadratic forms. A brief resume of pertinent aspects of this field will, therefore, be given in Chapter I. The reader is assumed to be familiar with the subject in general.

§1. Linear Equations and Linear Transformations

1. Vectors. The results of the theory of linear equations can be expressed concisely by the notation of vector analysis. A system of

real numbers $x_1,\ x_2,\ \cdots\ ,\ x_n$ is called an n-dimensional vector or a vector in n-dimensional space and denoted by the bold face letter $\mathrm{\vec{x}}$ ; the numbers $x_{i}\ \ \ (i=1,\ \cdots\ ,\ n)$ are called the components of the vector $\mathrm{\vec{x}}$ . If all components vanish, the vector is said to be zero or the null vector; for n=2

a vector can be interpreted geometricallyas a “position vector” leading from the origin to the point with the rectangular coordinates $x_{i}$ . For n>3

geometrical visualization is no longer possible but geometrical terminology remains suitable.

Given two arbitrary real numbers $\lambda$ and $\mu$ , the vector $\lambda\mathrm{\vec{x}}+\mu\mathrm{\vec{y}}=\mathrm{\vec{z}}$ is defined as the vector whose components $z_{i}$ are given by $z_{i}=\lambda{x}_{i}+\mu{y}_{i}$ . thus in particular, the sun and difference of two vectors are defined.

The number

(1) $\mathrm{\vec{x}}\cdot\mathrm{\vec{y}}=x_{1}y_{1}+\ \cdots\ +x_{n}y_{n}$ $=y_{1}x_{1}+\ \cdots\ y_{n}x_{n}=\mathrm{\vec{y}}\cdot\mathrm{\vec{y}}$

is called the “inner product” of the vectors $\mathrm{\vec{x}}$ and $\mathrm{\vec{y}}$ .

Occasionally we shall call the inner product $\mathrm{\vec{x}}\cdot\mathrm{\vec{y}}$ the component of the vector $\mathrm{\vec{y}}$ with respect to $\mathrm{\vec{x}}$ or vice versa.

If the inner product $\mathrm{\vec{x}}\cdot\mathrm{\vec{y}}$ vanishes we say that the vectors $\mathrm{\vec{x}}$ and $\mathrm{\vec{y}}$ are orthogonal ; for n=2

and

this terminology has an immediate geometrical meaning. The inner product $\mathrm{\vec{x}}\cdot\mathrm{\vec{y}}$ of a vector with itself plays a special role ; it is called the norm of the vector. The positive square root of $\mathrm{\vec{x}}^{2}$ is called the length of the vector and denote by $|\mathrm{\vec{x}}|=$ $\sqrt{\mathrm{\vec{x}}^{2}}$ . A vector whose length is unity is called a normalized vector or unit vector.

The following inequality is satisfied by the inner product of two vectors $\mathrm{\vec{a}}=(a_{1},\ \cdots\ a_{n})$ and $\mathrm{\vec{b}}=(b_{1},\ \cdots\ b_{n})$ :

$(\mathrm{\vec{a}}\cdot\mathrm{\vec{b}})^{2}\le\mathrm{\vec{a}}^{2}\mathrm{\vec{b}}^{2}$

or without using vector notation,

$\left(\sum_{i=1}^{n}a_{i}b_{i}\right)^{2}\le\left(\sum_{i=1}^{n}a_{i}^{2}\right)\left(\sum_{i=1}^{n}b_{i}^{2}\right)$ ,

where the equality holds if and only if the $a_{i}$ and the $b_{i}$ are proportional, i.e. if a relation of the form $\lambda\mathrm{\vec{a}}+\mu\mathrm{\vec{b}}=0$ with $\lambda^{2}+\mu^{2}\ne{0}$ is satisfied.

The proof of this “schwarz inequality” follows from the fact that the roots of the quadratic equation

$\sum_{i=1}^{n}(a_{i}x+b_{i})^2=x^{2}\sum_{i=1}^{n}a_{i}^{2}+2x\sum_{i=1}^{n}a_{i}b_{i}+\sum_{i=1}^{n}b_{i}^{2}=0$

for the unknown

can never be real and distinct, but must be imaginary, unless the $a_{i}$ and $b_{i}$ are proportional. The Schwarz inequality is merely an expression of this fact in terms of the discriminant of the equation. Another proof of the Schwarz inequality follows immediately from the identity

$\sum_{i=1}^{n}a_{i}^{2}\sum_{i=1}^{n}b_{i}^{2}-\left(\sum_{i=1}^{n}a_{i}b_{i}\right)^{2}$ $=\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}(a_{j}b_{k}-a_{k}b_{j})^{2}$ .

Vectors $\mathrm{\vec{x}_{1},\ \mathrm{\vec{x}_{2},\ \cdots\ ,\mathrm{\vec{x}_{m}$ are said to be linearly dependent if a set of numbers $\lambda_{1},\ \lambda{2},\ \cdots\ ,\ \lambda_{m}$ (not all equal to zero) exists such that the vector equation

$\lambda_{1}x_{1}+\ \cdots\ +\lambda_{m}x_{m}=0$

is satisfied, i.e. such that all the components of the vector on the left vanish. Otherwise the vectors are said to be linearly independent.

The

vectors $\mathrm{\vec{e}}_{1}\ ,\ \mathrm{\vec{e}_{2}}\ ,\ \cdots\ ,\ \mathrm{\vec{e}}_{n}$ in

-dimensional space whose components are given , respectively, by the first, second, and

-th rows of the array

$\begin{matrix}1&0&\cdots&0\ \ \\0&1&\cdots&0\ \ \\\\\\&\cdots&\cdots&\\\\\\0&0&\cdots&1\ ,\end{matrix}$

form a system of

linearly independent vectors.