= Image-kernel orthogonality for an adjoint
{title2=$\operatorname{im}T=(\ker T^*)^\perp$}
For a <linear map> $T:V\to W$ between finite-dimensional <inner-product space>[inner product spaces],
$$
(\operatorname{im}T)^\perp=\ker T^*,
\qquad
\operatorname{im}T=(\ker T^*)^\perp.
$$
The first equality follows directly from the defining identity for the <adjoint operator>; the second follows by taking <orthogonal complement>[orthogonal complements]. In an infinite-dimensional <Hilbert space>, the second equality generally requires closure of the image.
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