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Let $G$ be a linear Lie group acting properly and isometrically on a$G$-spin$^c$ manifold $M$ with compact quotient. We show that Poincar\'eduality holds between $G$-equivariant $K$-theory of $M$, defined usingfinite-dimensional $G$-vector bundles, and $G$-equivariant $K$-homology of $M$,defined through the geometric model of Baum and Douglas.

An equivariant Poincar\'e duality for proper cocompact actions by matrix groups