Question: Show that the terms 'nilpotent transformation' and 'nilpotent matrix', as given in Definition 2.7, fit with each other: a map is nilpotent if and only

Show that the terms 'nilpotent transformation' and 'nilpotent matrix', as given in Definition
2.7, fit with each other: a map is nilpotent if and only if it is represented by a nilpotent matrix. (Is it that a transformation is nilpotent if an only if there is a basis such that the map's representation with respect to that basis is a nilpotent matrix, or that any representation is a nilpotent matrix?)

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