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Brown Corpus
We are trying to study a linear operator T on the finite-dimensional space V, by decomposing T into a direct sum of operators which are in some sense elementary.
We can do this through the characteristic values and vectors of T in certain special cases, i.e., when the minimal polynomial for T factors over the scalar field F into a product of distinct monic polynomials of degree 1.
Second, even if the characteristic polynomial factors completely over F into a product of polynomials of degree 1, there may not be enough characteristic vectors for T to span the space V.
The second situation is illustrated by the operator T on Af ( F any field ) represented in the standard basis by Af.
The characteristic polynomial for A is Af and this is plainly also the minimal polynomial for A ( or for T ).
On the other hand, the null space of Af and the null space of Af together span V, the former being the subspace spanned by Af and the latter the subspace spanned by Af and Af.
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