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Matrices of Linear Transformations from a Basis of a Subspace to an Extended Basis 📂Linear Algebra

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█