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Mathematics-Online course: Linear Algebra - Matrices - Linear Maps

Matrix

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A $ (m\times n)$-matrix over field $ K$ is a set of numbers from $ K$ arranged in a rectangular grid as follows:

$\displaystyle A = \left( \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & \cd... ... a_{m3} & \cdots & a_{mn} \end{array} \right) \qquad n,m \in \mathbb{N} $

Abbreviated form: $ A=(a_{ij})$

$ (a_{i1},a_{i2},\dots,a_{in})$ is called $ i$-th row vector of $ A$.

$ (a_{1j},a_{2j},\dots,a_{mj})$ is called $ j$-th column vector of $ A$.

Observe: 1st index = row index
  2nd index = column index
A $ (1 \times m)$-matrix is called row vector.

A $ (n \times 1)$-matrix is called column vector.

Row vectors as well as column vectors can be considered as elements of $ K^n$ and vice versa. The set of all $ (n\times m)$-matrices is denoted by $ K^{n \times m}$.

According to the above remark we have

$\displaystyle \begin{array}{lcl} K^{1 \times m} & = & K^m \\ K^{n \times 1} & = & K^n \\ K^{1 \times 1} & = & K \end{array} $

The following figure shows some real and complex matrices.

$\displaystyle \left( \begin{array}{c} 31\\ 57\\ 97\end{array}\right)\,,\quad \... ...5 \\ 21 & 22 & 23 & 24 & 25 \\ 31 & 32 & 33 & 34 & 35 \end{array}\right) $

The first two examples represent column and row vectors, resp., that is, these are matrices one dimension of which equals $ 1$. On the right hand side you can see a square and a rectangular matrix.

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005