Matrices of Linear Transformations from a Basis of a Subspace to an Extended Basis
Theorem
Let █eq01█ be a subspace in █eq02█ dimension vector space called █eq03█. Let █eq04█ be the ordered basis of █eq05█. Consider █eq06█ as an extension of the basis of █eq03█ from █eq07█. Let █eq09█ be a linear transformation. Then, the matrix representation of █eq11█ for █eq10█ is the following block matrix:
█eq1█
Here, █eq12█, █eq13█ is a contraction map, █eq14█ is a matrix of █eq15█, █eq16█ is a matrix of █eq17█, ██eq18██ is an █eq19█ zero matrix.
Proof
Let █eq13█ be a contraction map. Let █eq21█.
█eq2█
To find the matrix representation, one needs to know how the elements of the basis are mapped. In other words, the component █eq23█ in the first column of █eq22█ is the same as the coefficient of █eq26█ when expressing █eq24█ as a linear combination of █eq10█. But since █eq27█,
█eq3█
Therefore, █eq28█. Thus, when █eq29█, █eq30█, and when █eq31█, █eq32█.
█eq4█
By following the same method for the components up to the █eq33█-th column,
█eq5█
In this case, if █eq34█, and assuming █eq35█, █eq36█, then
█eq6█
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