Matrix/K/Transposed matrix/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}M=(a_{ij})_{ij}}$ be an ${\displaystyle {}m\times n}$-matrix over ${\displaystyle {}K}$. Then the ${\displaystyle {}n\times m}$-matrix

${\displaystyle {M^{\text{tr}}}={\left(b_{ij}\right)}_{ij}{\text{ with }}b_{ij}:=a_{ji}}$

is called the transposed matrix of ${\displaystyle {}M}$.