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Mathematics-Online course: Preparatory Course Mathematics - Linear Algebra and Geometry - Systems of Linear Equations


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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 =
a_{11} & a_{12} & a_{13} & \cd...
... a_{m3} & \cdots & a_{mn}
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

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  automatically generated 1/9/2017