Introduction of Vector Spaces

VECTOR SPACES

INTRODUCTION

Many physical phenomena such as forces, velocities and accelerations involve both magnitude and direction. Such entity involving both magnitude and direction is known as vector.

A vector is denoted by an arrow whose length denotes the magnitude of the vector and whose direction represents the direction of the vector. In most of the physical situations in the real life, involving vectors, only the magnitude and direction of the vector are significant; consequently, we regard vectors with the same magnitude and direction as being equal irrespective of their positions.

The vectors used to represent the physical quantities can be combined to form a resultant vector that represents the combined effects of the original quantities. This resultant vector is called the sum of the original vectors, and the rule for their combination is called the parallelogram law.

Parallelogram law for vector addition

The sum of two vectors x and y that act at the same point A is the vector beginning at A that is represented by the diagonal of parallelogram having x and y as adjacent sides.

Besides the operation of vector addition, the another operation performed on vectors is the length of a vector may be magnified or contracted. This operation is called scalar multiplication. It consists of multiplication of a vector with a real number. If the vector x is represented by an arrow, then for any real number t, the vector tx is represented by an arrow in the same direction tx is |t| times the length of the arrow x. The non−zero vectors x and y are called parallel if y=tx for same non−zero real number t. Thus non−zero vectors having the same or opposite directions are parallel.

Properties

(i) For all vectors x and y, x+y=y+x.

(ii) For all vectors x, y and z, (x + y) + z = x + (y + z).

(iii) There exists a vector denoted by 0 such that x+0=x, for each vector x.

(iv) For each vector x, there is a vector y such that x+y=0

(v) For each vector x, 1.x = x

(vi) For each pair of real numbers a and b each vector x, (ab) x=a (bx)

(vii) For each real number a and each pair of vectors x and y, a (x + y) = ax + ay.

(viii) For each pair of real numbers a and b and each vector x, (a+b) x = ax + bx.

Note: Any mathematical structure possessing the above 8 properties is called a vector space.