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In this paper we classify a linear family of Lie brackets on the space ofrectangular matrices $Mat(n\times m,\K)$ and we give an analogue of the Ado'sTheorem. We give also a similar classification on the algebra of the squarematrices $Mat(n, \K)$ and as a consequence, we prove that we can't built afaithful representation of the $(2n+1)$-dimensional Heisenberg Lie algebra$\mathfrak{H}_n$ in a vector space $V$ with $\dim V\leq n+1$. Finally, we provethat in the case of the algebra of square matrices $Mat(n,\K)$, thecorresponding Lie algebras structures are a contraction of the canonical Liealgebra $\mathfrak{gl}(n,\K)$.

Linear family of Lie brackets on the space of matrices $Mat(n\times m,\K)$ and Ado's Theorem