Let K {\displaystyle {}K} denote a field, and let V {\displaystyle {}V} denote a K {\displaystyle {}K} -vector space of dimension n {\displaystyle {}n} . Let v = v 1 , … , v n {\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}} and w = w 1 , … , w n {\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{n}} denote two bases of V {\displaystyle {}V} . Let
with coefficients c i j ∈ K {\displaystyle {}c_{ij}\in K} . Then the n × n {\displaystyle {}n\times n} -matrix
is called the transformation matrix of the base change from v {\displaystyle {}{\mathfrak {v}}} to w {\displaystyle {}{\mathfrak {w}}} .
denote a
field,
and let 
denote a
-vector space
of
dimension
. Let
and 
denote two
bases
of . Let
.
Then the
-matrix
to 
.
