This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let λi be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in Rn, with Neumann homogeneous boundary conditions on Γ = ∂Ω. Let {φij} j=1 be the corresponding linearly independent (normalized) eigenfunctions in L2(Ω), so that `i is the geometric multiplicity of λi. We prove that the Dirichlet boundary traces {φij |Γ1} `i j=1 are linearly independent in L2(Γ1). Here Γ1 is an arbitrary open, connected portion of Γ , of positive surface measure. The same conclusion holds true if the setting {Neumann B.C., Dirichlet boundary traces} is replaced by the setting {Dirichlet B.C., Neumann boundary traces}. These results are motivated by boundary feedback stabilization problems for parabolic equations [L-T.2]. Part II: The same problem is posed for the Stokes operator with motivation coming from the boundary stabilization problems in [B-L-T.1]– [B-L-T.3] (with tangential boundary control), and [R] (with just boundary control), where we take Γ1 = Γ . The aforementioned property of boundary traces of eigenfunctions critically hinges on a unique continuation result from the boundary of corresponding over-determined problems. This is well known in the case of second-order elliptic operators of Part I; but needs to be established in the case of Stokes operators. A few proofs are given here. 2000 Mathematics Subject Classification: 35L20, 47, 49K20, 76N25, 76Q05, 93B29, 93C20.