Let
be a
field,
and let
be a
-vector space
of
dimension
. Let
and
denote
bases
of
. Suppose that
-
![{\displaystyle {}v_{j}=\sum _{i=1}^{n}c_{ij}w_{i}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6118e8c8ff2013e83a6d267e4aaa890cf4cc8171)
with coefficients
,
which we collect into the
-matrix
-
![{\displaystyle {}M_{\mathfrak {w}}^{\mathfrak {v}}={\left(c_{ij}\right)}_{ij}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a53b930e64465f0f0df72e0ae9cc02eaecb0396)
Then a vector
![{\displaystyle {}u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40d2bf6b1b7dfd0fca27756af08c9c7b277fc080)
, which has the coordinates
![{\displaystyle {}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5125fe8d9fb09fcbaadabfc066b66d966910b853)
with respect to the basis
![{\displaystyle {}{\mathfrak {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/141620e29cf8517dee128b1cf63c7226b9d95872)
, has the coordinates
-
![{\displaystyle {}{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}=M_{\mathfrak {w}}^{\mathfrak {v}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}={\begin{pmatrix}c_{11}&c_{12}&\ldots &c_{1n}\\c_{21}&c_{22}&\ldots &c_{2n}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\ldots &c_{nn}\end{pmatrix}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be80435d7fc625371c90efd7b7f5ebda1909717)
with respect to the basis
![{\displaystyle {}{\mathfrak {w}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17817534496a744d36ead0f08241c66070b09982)
.