# Linear mapping/Matrix/Relation/Section

Due to fact, a linear mapping

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.

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

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*.

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.

Let be a field, and let be an -dimensional vector space with a basis , and let be an -dimensional vector space with a basis . Then the mappings

defined in definition, are inverse to each other.

### Proof

is usually described by the matrix with respect to the standard bases on the left and on the right. The result of the matrix multiplication

can be interpreted directly as a point in . The -th column of is the image of the -th standard vector .