Editing Index notation (section)

===One-dimensional arrays (vectors)===

{{main|Vector (mathematics and physics)}}

A vector treated as an array of numbers by writing as a [[row vector]] or [[column vector]] (whichever is used depends on convenience or context):

:<math>\mathbf{a} = \begin{pmatrix}
    a_1 \\
    a_2 \\
    \vdots \\
    a_n  
  \end{pmatrix}, \quad \mathbf{a} = \begin{pmatrix}
    a_1 & a_2 & \cdots & a_n
\end{pmatrix}</math>

Index notation allows indication of the elements of the array by simply writing ''a<sub>i</sub>'', where the index ''i'' is known to run from 1 to ''n'', because of n-dimensions.<ref>An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, {{ISBN|0-582-44355-5}}</ref>
For example, given the vector:

:<math>\mathbf{a} = \begin{pmatrix}
  10 & 8 & 9 & 6 & 3 & 5 \\ 
\end{pmatrix}</math>

then some entries are

:<math>a_1 = 10,\, a_2 = 8,\, \cdots,\, a_6 = 5 </math>.

The notation can be applied to [[vectors in mathematics and physics]]. The following [[vector equation]]

:<math>\mathbf{a} + \mathbf{b} = \mathbf{c}</math>

can also be written in terms of the elements of the vector (aka components), that is

:<math> a_i + b_i = c_i </math>

where the indices take a given range of values. This expression represents a set of equations, one for each index. If the vectors each have ''n'' elements, meaning ''i'' = 1,2,…''n'', then the equations are explicitly

:<math>\begin{align}
  a_1 + b_1 &= c_1 \\
  a_2 + b_2 &= c_2 \\
            &\ \ \vdots \\
  a_n + b_n &= c_n
\end{align}</math>

Hence, index notation serves as an efficient shorthand for
#representing the general structure to an equation,
#while applicable to individual components.