Column of a Matrix

Definition

A straight vertical path in a matrix to arrange the elements in the same path is called a column of a matrix.

columns of matrix

The internal space in a matrix is divided into number of columns as per the following two factors.

  1. The total number of elements.
  2. The arrangement of number of elements equally in each column.

The elements are arranged in vertical straight path but they are separated by some space.

The arrangement of elements in columns can be expressed in general form as follows.

$ \begin{bmatrix} \begin{array}{|c|c|c|c|c|} \hline e_{11} & e_{12} & e_{13} & \cdots & e_{1n} \\ e_{21} & e_{22} & e_{23} & \cdots & e_{2n} \\ e_{31} & e_{32} & e_{33} & \cdots & e_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ e_{m1} & e_{m2} & e_{m3} & \cdots & e_{mn} \\\hline \end{array} \end{bmatrix} $

In this general form matrix, the total number of elements are $mn$. The internal space of a matrix is split into $n$ columns to place $m$ elements in each column.

Closely observe each element in the matrix and every element is represented by two numbers at its subscript position. The second number represents the number of the respective column. For example, the second number of every element in subscript position for first column is $1$. It is $2$ for second column, $3$ for third column and continued till last column. The last column is $n^{th}$ column. Therefore, the second number in subscript position of each element in last column is $n$.

The general form matrix is expressed in compact form as follows.

$ \begin{bmatrix} e_{\displaystyle ij} \end{bmatrix} $

$e_{ij}$ is an element in general form and it represents every element in the matrix. The second letter in subscript position of this element is $j$ and it represents the number of respective column.

Example

The following examples are example column matrices for your understanding.

$ \begin{bmatrix} \begin{array}{|c|} \hline 8 \\\hline \end{array} \end{bmatrix} $

$8$ is only one element in matrix. It is placed in one column. Therefore, the element $e_{11} = 8$.

$ \begin{bmatrix} \begin{array}{|c|} \hline 6 \\ -2 \\ 7 \\\hline \end{array} \end{bmatrix} $

$6$, $-2$ and $7$ are elements and they are placed only in one column. So, the elements are $e_{11} = 6$, $e_{21} = -2$ and $e_{31} = 7$.

$ \begin{bmatrix} \begin{array}{|c|c|} \hline 5 & 0 \\ 4 & -3 \\ 1 & 9 \\ 8 & 6 \\\hline \end{array} \end{bmatrix} $

In this example, eight elements are arranged in two columns. So, each four elements are placed in one column. Therefore, the elements $e_{11} = 5$, $e_{21} = 4$, $e_{31} = 1$, $e_{41} = 8$, $e_{12} = 0$, $e_{22} = -3$, $e_{32} = 9$ and $e_{42} = 6$.

$ \begin{bmatrix} \begin{array}{|c|c|c|c|} \hline -9 & -6 & 2 & 4 \\ 8 & -7 & 5 & 2 \\ 9 & 4 & -3 & 5 \\\hline \end{array} \end{bmatrix} $

The twelve elements are arranged in four columns and each column contains three elements. So, $e_{11} = -9$, $e_{21} = 8$, $e_{31} = 9$, $e_{12} = -6$, $e_{22} = -7$, $e_{32} = 4$, $e_{13} = 2$, $e_{23} = 5$, $e_{33} = -3$, $e_{14} = 4$, $e_{24} = 2$ and $e_{34} = 5$.


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