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Section 45.2 Terminology and notation
The first two definitions below apply to a linear transformation \(\funcdef{T}{V}{W}\) between finite-dimensional vector spaces \(V\) and \(W\text{,}\) along with some choice of bases \(\basisfont{B}\) and \(\basisfont{B}'\) of \(V\) and \(W\text{,}\) respectively.
matrix of \(T\) relative to \(\basisfont{B}, \basisfont{B}'\)
the standard matrix of the composition
\(\invcoordmap{B} T \coordmap{B'}\text{,}\) which is a transformation
\(\R^n \to \R^m\) in the real case or a transformation
\(\C^n \to \C^m\) in the complex case, and where
\(n = \dim V\) and
\(m = \dim W\)
\(\matrixOf{T}{B'B}\)
notation for the
matrix of \(T\) relative to \(\basisfont{B}\) and \(\basisfont{B}'\)
The next two definitions apply to a linear operator \(\funcdef{T}{V}{V}\text{.}\) In the first, \(V\) is assumed to be finite-dimensional, and \(\basisfont{B}\) is some basis of \(V\) that is chosen to represent both a basis for the domain space of \(T\) and the codomain space of \(T\) (which are the same space in this case).
\(\matrixOf{T}{B}\)
notation for the
matrix of \(T\) relative to \(\basisfont{B}\)
\(T^\ell\)
notation for composition of
\(\abs{\ell}\) copies of either
\(T\) (
\(\ell\) positive) or
\(\inv{T}\) (
\(\ell\) negative)
