This paper deals with (1) the truncated matrix Hamburger moment problem from the point of view of reproducing kernel Hilbert spaces of vector valued entire functions of the kind introduced and extensively studied by Louis de Branges and (2) the truncated matrix trigonometric moment problem viewed through an analogous class of spaces that are formulated with respect to the open unit disc rather than the open upper half-plane. In this approach projections are computed via appropriately chosen reproducing kernels instead of orthogonal bases. This approach eases the bookkeeping and leads to pleasing formulas.