# Nilpotent transformation

Consider the transformation N: V->V. Let g be a vector such that N^k-1 does not equal 0, but N^k = 0. First show that the vectors g,N(g),N^2(g),..,N^k-1(g) are linearly independent, and then (assuming V has dimension n) If N is nilpotent of index n, show that the set S= {g, N(g), N^2(g),...,N^n-1(g)}is a basis for V. Describe the matrix which represents N with respect to the basis S.

Â© BrainMass Inc. brainmass.com February 24, 2021, 2:20 pm ad1c9bdddfhttps://brainmass.com/math/linear-transformation/nilpotent-transformation-14775

#### Solution Preview

Please see the attachment.

Consider the transformation N: V->V. Let g be a vector such that . First show that the vectors are linearly independent, and then (assuming V has dimension n) If N is nilpotent of index n, show that the set

S= { }is a basis for V. Describe the matrix which represents N with respect to the basis S.

Statement 1: Consider the transformation N: V->V. Let g ...

#### Solution Summary

This shows how to show vectors are linearly independent, that a given set is a basis for a vector, and describes the matrix for the basis.