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Definition of Vectors 📂Matrix Algebra

Definition of Vectors

Definition

A sequence of numbers is called a vector.

Description

In the general curriculum, a vector is learned as a ‘geometric object with magnitude and direction’. Since it’s the concept you first come across in physics, you inevitably become familiar with vectors of 33 dimensions or less.

(3,4)=[34] (3,4) = \begin{bmatrix} 3 \\ 4 \end{bmatrix} (x,y,z)=[xyz] (x,y,z) = \begin{bmatrix} x \\ y \\ z \end{bmatrix}

However, vectors can actually be generalized to more coordinates. It’s sufficient to just list more numbers below, for example, a 44-dimensional vector considering time tt can be denoted as follows.

(t,x,y,z)=[txyz] (t,x,y,z) = \begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix}

What does it mean to have vectors of more than 44 dimensions? For instance, if you want to represent the position (x,y,z)(x,y,z) at time tt and the thermal energy EE for each oxygen molecule, you can extend it to 55 dimensions as follows.

(t,x,y,z,E)=[txyzE] (t,x,y,z,E) = \begin{bmatrix} t \\ x \\ y \\ z \\ E \end{bmatrix}

The point is, there’s no need to be overly afraid of the length of the vector, that is, the increase in dimensions. In the endless world of mathematics given under a specific format, such expansion of dimensions is natural and obvious. In the same manner, it’s possible to think of vectors generalized up to nn dimensions, usually denoted by boldface x\mathbf{x}.

x=(x1,,xn)=[x1xn] \mathbf{x} = \left( x_{1}, \cdots , x_{n} \right) = \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix}

From this simple definition, a nn-dimensional vector is indistinguishable from a nn-tuple. The farther you get from physics and closer to mathematics, the less you use expressions like x\vec{x} with arrows, and as you enter into abstract and general mathematics, you get precise and strict definitions without terms like ‘coordinates’ or ‘sequence’.

See Also