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Is it ok to assume that $\operatorname{dim}(\operatorname{Im}(T^*))=\operatorname{dim}[(\operatorname{Im}(T))^*]$, where $T$ is a linear map acting on a finite dimensional space. i.e. just taking the dual outside?

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    Is your space finite dimensional? If so, the dimension of the dual equals the dimension of the original space, and the dimension of the image of $T^*$ is the rank of any matrix representation of $T^*$, which is the rank of the transpose conjugate of a matrix representation of $T$, which has the same rank as any matrix representation of $T$.2012-05-22
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    @ArturoMagidin: Thank you! I never knew that interpretation in terms of matrices.2012-05-22

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