# Vector Spaces

Giusto Bellavitis | Michail Ostrogradsky | William Hamilton |

In applications, people deal with a variety of quantities that are used to describe the physical world. Examples of
such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work,
power, etc. All these quantities can by divided into two categories -- vectors and scalars. A vector quantity is a
quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity
that is fully described by its magnitude.
In mathematics, physics, and engineering, a Euclidean **vector**
(simply a vector) is a geometric object
that has magnitude (or length) and direction. Many familiar physical notions, such as forces, velocities, and
accelerations, involve both magnitude (the amount of the force, velocity, or acceleration) and a direction. In most
physical situations involving vectors, only the magnitude and direction of the vector are significant; consequently,
we regard vectors with the same length and direction as being equal irrespective to their positions.

It is a custom to identify vectors with arrows (geometric object). The tail of the arrow is called the
**initial point** of the vector and the tip
the **terminal point**. To emphasis this approach, an arrow is placed above the initial and terminal
points, for example, the notation \( {\bf v} = \vec{AB} \) tells us that *A* is the
starting point of the vector **v** and its terminal point is *B*. In this tutorial
(as in most science papers and textbooks), we will denote vectors in boldface type applied to lower case
letters such as **v**, **u**, or **x**.

Any two vectors **x** and **y** can be added in "tail-to-head" manner; that is, either
**x** or **y** may be applied to any point and then another vector is applied to the
endpoint of the first. If this is done, the endpoint of the latter is the endpoint of their sum, which is denoted
by **x** + **y**. Besides the operation of vector addition there is another natural
operation that can be performed on vectors---multiplication by a scalar that are often taken to be real numbers.
When a vector is multiplied by a real number *k*, its magnitude is multiplied by |*k*| and its direction
remains the same when *k* is positive and the opposite direction when *k* is negative. Such vector is
denoted by *k***x**.

The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. The Italian
mathematician, senator, and municipal councilor Giusto Bellavitis (1803--1880) abstracted the basic idea in 1835.
The idea of an *n*-dimensional Euclidean space for *n* > 3 appeared in a work on the divergence theorem
by the Russian mathematician Michail Ostrogradsky (1801--1862) in 1836, in the geometrical tracts of Hermann Grassmann (1809--1877) in the early 1840s,
and in a brief paper of Arthur Cayley (1821--1895) in 1846. Unfortunately, the first two authors were virtually ignored in their lifetimes.
In particular, the work of Grassmann was quite philosophical and extremely difficult to read.
The term vector was introduced by the Irish mathematician, astronomer, and mathematical physicist William Rowan
Hamilton (1805--1865) as part of a quaternion.

Vectors can be described also algebraically. Historically, the first vectors were Euclidean vectors that can be
expanded through standard basic vectors that are used as coordinates. Then any vector can be uniquely represented
by a sequence of scalars called coordinates or components. The set of such ordered *n*-tuples is denoted by
\( \mathbb{R}^n . \) When scalars are complex numbers, the set of ordered *n*-tuples
of complex numbers is denoted by \( \mathbb{C}^n . \) Motivated by these two approaches, we
present the general definition of vectors.