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    Range, rank, null space, and nullity of the zero and identity operators

    E-HK-31-2

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

    Find the range, rank, null space, and nullity for the zero transformation and the identity transformation on a finite-dimensional space \(V\).

    Solution

    Let \(\dim V=n\).

    For the zero transformation \(Z(v)=0\): \[ \operatorname{range}Z=\{0\}, \qquad \operatorname{rank}Z=0, \] \[ \ker Z=V, \qquad \operatorname{nullity}Z=n. \]

    For the identity transformation \(I(v)=v\): \[ \operatorname{range}I=V, \qquad \operatorname{rank}I=n, \] \[ \ker I=\{0\}, \qquad \operatorname{nullity}I=0. \]

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