Linear Transformation

In line with the modern mathematical point-of-view, we adopt the coordinate-free point-of-view for linear algebra. A vector in a Euclidean space exists independently of any coordinates used to describe it. A linear transformation from one Euclidean space to another maps one vector to another, independently of any coordinate system. In contrast, in matrix algebra, one works with a basis. A matrix describes how a linear transformation maps one vector to another. In economic theory and econometrics, typically vectors are not seen as coordinate-free. A particular basis is singled out, and one works with coordinates. Despite this tradition, the coordinate-free point-of-view is superior. Not using coordinates reduces the use of subscripts and makes expressions simpler, and theorems are easier to state and to prove. Given a linear transformation A:X→Y, then there exists a unique linear transformation (the adjoint) A⊤:Y→X that preserves the inner product: $$\left \langle \boldsymbol {y}^{\ast },{\textit {{\sf { {A}}}}}\boldsymbol {x}\right \rangle =\left \langle {\textit {{\sf { {A}}}}}^{\top }\boldsymbol {y}^{\ast },\boldsymbol {x}\right \rangle \!.$$ Let R(·) and N(·) denote the range and the null space of a linear transformation. The “fundamental theorem of linear algebra” asserts that the orthogonal complement of the null space of a linear transformation is the range of the adjoint. Given a linear transformation A:X→Y, the restricted linear transformation $${\textit {{\sf { {A}}}}}:\mathrm {R}\left ( {\textit {{\sf { {A}}}}}^{\top }\right ) \rightarrow \mathrm {R}\left ( {\textit {{\sf { {A}}}}}\right )$$ is invertible. Extending the inverse of the restricted linear transformation obtains the Moore-Penrose generalized inverse A+:Y→X: for $$\boldsymbol {y}\in \mathrm {R}\left ( {\textit {{\sf { {A}}}}}\right )$$ , then A+y is the inverse of the restricted linear transformation; for $$\boldsymbol {y}\in \mathrm {N}\left ( {\textit {{\sf { {A}}}}}^{\top }\right )$$ , then A+y=0.