## Linear Transformation and a Basis of the Vector Space $\R^3$

## Problem 182

Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$.

Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$.

Show that the vectors $\mathbf{x}, T\mathbf{x}, T^2\mathbf{x}$ form a basis for $\R^3$.

(*The Ohio State University Linear Algebra Exam Problem*)

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