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Vector Space Example 2Example II: Let M be the set of all (m×n) matrices, with complex numbers as entries. Let C be the field of complex numbers. Then if -
p_{11} & p_{12} & p_{13} & \cdots & p_{1n} \\ p_{21} & p_{22} & p_{23} & \cdots & p_{2n} \\ p_{31} & p_{32} & p_{33} & \cdots & p_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{m1} & p_{m2} & p_{m3} & \cdots & p_{mn} \end{bmatrix} Define vector addition in M: -
P+Q & = & \begin{bmatrix} p_{11} & p_{12} & p_{13} & \cdots & p_{1n} \\ p_{21} & p_{22} & p_{23} & \cdots & p_{2n} \\ p_{31} & p_{32} & p_{33} & \cdots & p_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{m1} & p_{m2} & p_{m3} & \cdots & p_{mn} \end{bmatrix} + \begin{bmatrix} q_{11} & q_{12} & q_{13} & \cdots & q_{1n} \\ q_{21} & q_{22} & q_{23} & \cdots & q_{2n} \\ q_{31} & q_{32} & q_{33} & \cdots & q_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ q_{m1} & q_{m2} & q_{m3} & \cdots & q_{mn} \end{bmatrix} \\ & = & \begin{bmatrix} p_{11}+q_{11} & p_{12}+q_{12} & p_{13}+q_{13} & \cdots & p_{1n}+q_{1n} \\ p_{21}+q_{21} & p_{22}+q_{22} & p_{23}+q_{23} & \cdots & p_{2n}+q_{2n} \\ p_{31}+q_{31} & p_{32}+q_{32} & p_{33}+q_{33} & \cdots & p_{3n}+q_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{m1}+q_{m1} & p_{m2}+q_{m2} & p_{m3}+q_{m3} & \cdots & p_{mn}+q_{mn} \end{bmatrix} \end{matrix} Define scalar multiplication: -
p_{11} & p_{12} & p_{13} & \cdots & p_{1n} \\ p_{21} & p_{22} & p_{23} & \cdots & p_{2n} \\ p_{31} & p_{32} & p_{33} & \cdots & p_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{m1} & p_{m2} & p_{m3} & \cdots & p_{mn} \end{bmatrix} = \begin{bmatrix} c \cdot p_{11} & c \cdot p_{12} & c \cdot p_{13} & \cdots & c \cdot p_{1n} \\ c \cdot p_{21} & c \cdot p_{22} & c \cdot p_{23} & \cdots & c \cdot p_{2n} \\ c \cdot p_{31} & c \cdot p_{32} & c \cdot p_{33} & \cdots & c \cdot p_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c \cdot p_{m1} & c \cdot p_{m2} & c \cdot p_{m3} & \cdots & c \cdot p_{mn} \end{bmatrix} Then M is a vector space over C and we denote this as Cm×n. So Example I would be denoted R1×n, or more simply, Rn. In analysis, many function sets have the structure of a vector space. In analysis, a vector space is called a linear space. - See also: vector space.
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