## Cours de mathématiques Tome 4 Equations différentielles by J.Lelong-Ferrand, J.M.Arnaudiès

By J.Lelong-Ferrand, J.M.Arnaudiès

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Basis of a vector space) A linearly independent set B of vectors is said to be a (Hamel) basis of V if every vector of V is a linear combination of the vectors in B. The vector space V is said to be finite dimensional if there exists a Hamel basis B consisting of finitely many vectors. It is not clear at the outset whether a vector space possesses a basis. Using Zorn's lemma, one can demonstrate the existence of a maximal linearly independent system of vectors in any vector space. ) Any maximal set is indeed a basis of the vector space.

Since Xl, X2 E M(X3, X4), it follows that M(xI, X2) = M(X3, X4). Consequently, there is one and only one line containing any two points. Consider now two different two-dimensional subspaces M (Xl, X2) and M(X3, X4) of V(F). The vectors x}, X2, X3 and X4 are linearly dependent as our vector space is only three dimensional. ) Obviously, y is non-zero. Clearly, the point y belongs to both the lines M(XI' X2) and M(X3, X4). This means that there is at least one point common to any two distinct lines.

8 1 n 8 2 = {a}. The following results give some properties of the operation + defined for subspaces. 7 Let 8 1 and 8 2 be two subspaces of a vector space V. Let 8 be the smallest subspace of V containing both 8 1 and 8 2 . Then (1) S = 8 1 + 82, (2) dim(S) = dim(8 1 ) + dim(S2) that 8 1 + 8 2 S dim(Sl n 8 2), PROOF. It is clear 8. Note that 8 1 + 8 2 is a subspace of V containing both 8 1 and 8 2. Consequently, 8 S 8 1 + S2. This establishes (1). • ,X r be a basis for 8 1 n 8 2, where r = dim(SI n 8 2).