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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 $3$ dimensions or less.

$$ (3,4) = \begin{bmatrix} 3 \\ 4 \end{bmatrix} $$ $$ (x,y,z) = \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$

However, vectors can actually be generalized to more coordinates. It’s sufficient to just list more numbers below, for example, a $4$-dimensional vector considering time $t$ can be denoted as follows.

$$ (t,x,y,z) = \begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix} $$

What does it mean to have vectors of more than $4$ dimensions? For instance, if you want to represent the position $(x,y,z)$ at time $t$ and the thermal energy $E$ for each oxygen molecule, you can extend it to $5$ dimensions as follows.

$$ (t,x,y,z,E) = \begin{bmatrix} t \\ x \\ y \\ z \\ E \end{bmatrix} $$

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 $n$ dimensions, usually denoted by boldface $\mathbf{x}$.

$$ \mathbf{x} = \left( x_{1}, \cdots , x_{n} \right) = \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix} $$

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

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