The effect of several linear mappings from to itself, represented on a brain cell.
is determined by the images
, ,
of the standard vectors. Every is a linear combination
and therefore the linear mapping is determined by the elements . So, such a linear map is determined by the elements
, , ,
from the field. We can write such a data set as a matrix. Because of
the determination theorem,
this holds for linear maps in general, as soon as in both vector spaces bases are fixed.
is called the linear mapping determined by the matrix.
For a linear mapping
,
we always assume that everything is with respect to the standard bases, unless otherwise stated. For a linear mapping
from a vector space in itself
(what is called an endomorphism),
one usually takes the same bases on both sides. The identity on a vector space of dimension is described by the identity matrix, with respect to every basis.