Matrix for linear mapping
Let
denote a
field,
and let
be an
-dimensional vector space
with a
basis
,
and let
be an
-dimensional vector space with a basis
.
For a
linear mapping
-
the
matrix
-
![{\displaystyle {}M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )=(a_{ij})_{ij}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/640a4d1dbe8ba74eb8430646a31ae8e1e461e8e4)
where
is the
-th
coordinate
of
with respect to the basis
, is called the describing matrix for
with respect to the bases.
For a matrix
,
the linear mapping
determined by
-
in the sense of
fact,
is called the linear mapping determined by the matrix
.