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Dirichlet's theorem in Diophantine approximation implies that for any real x, there exists a rational p/q arbitrarily close to x such that |x-p/q|<1/q^2. In addition, the exponent 2 that appears in this inequality is optimal, as seen for example by taking x=\sqrt{2}. In 1967, Wolfgang Schmidt suggested a similar problem, where x is a real subspace of R^d of dimension l, which one seeks to approximate by a rational subspace v. Our first goal will be to obtain the optimal value of the exponent in the analogue of Dirichlet's theorem within this framework. The proof is based on a study of diagonal orbits in the space of lattices in R^d. We shall also discuss other applications of our method, such as generalizations of Roth's theorem for Grassmann varieties, giving a formula for the Diophantine exponent of a linear subspace defined over a number field, or of Khintchine's theorem, which describes the Diophantine properties of points chosen randomly according to the Lebesgue measure.