# Data as a Linear Transformation ### Key Idea A linear transformation tells you *exactly* how one [space](Space.md) is transformed into another space, where a certain set of properties (i.e. linearity) is preserved. It can be represented as a matrix. This *representation* is with respect to a certain basis; particularly, the basis of the domain and codomain. If we were to change the basis, our matrix would of course change. However, the underlying *transformation* would *not change*! This is incredibly important. The transformation is *invariant* to the basis. Note, this is often referred to as a *parameterization* (see Betancourt). ### Misc See this notion doc: https://www.notion.so/Matrix-Transpose-SVD-Linear-Reg-Col-Space-3419582af5884d9f89638b52403d10ae I also touch on this [Data Matrices and Linear Transformation](Data%20Matrices%20and%20Linear%20Transformation.md) here, as well as in `Basis and data matrices` in notability. --- Date: 20220107 Links to: [Linear Algebra MOC](Linear%20Algebra%20MOC.md) Tags: References: * []()