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    $T^2 = 0$ relates range to null space; a square-zero example on $\RR^2$

    E-HK-32-8

    Area
    Algebra
    Topics
    Linear Algebra, Linear Transformations, Range and Null Space
    Status
    draft

    Let \(V\) be a vector space over the field \(F\) and \(T\) a linear operator on \(V\). If \(T^2 = 0\), what can you say about the relation of the range of \(T\) to the null space of \(T\)? Give an example of a linear operator \(T\) on \(\RR^2\) such that \(T^2 = 0\) but \(T \neq 0\).

    Solution

    If \(T^2=0\) and \(y\in\operatorname{range}T\), then \(y=T\alpha\) for some \(\alpha\), and \[ T(y)=T(T\alpha)=0. \] Thus every vector in the range of \(T\) lies in the null space: \[ \operatorname{range}T\subseteq\ker T. \]

    For example, on \(\mathbb R^2\) define \[ T(x,y)=(y,0). \] Then \(T\ne0\), since \(T(0,1)=(1,0)\), while \[ T^2(x,y)=T(y,0)=(0,0). \] Hence \(T^2=0\) but \(T\ne0\).

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