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A1, xn = bl, aml x1 + am2 X2 + ... + amnxn = bm, in which the coefficients ai; and the constant terms bi are fixed elements of F. The system is said to be homogeneous when b1= b2 = ... = bm = o. , xn) whose components satisfy all the relations. The definitions given above can be generalised by replacing the field F by a ring K. We cannot then, however, make use of the properties of vector spaces in the ensuing theory. Equations (not necessarily linear) over the ring Z are historically known as Diophantine equations.

This vector space is denoted by Hom(U, V), £°F(U, V) or £(U, V). , vm be a basis for V. , un of U. We have xl = allxl+a12x2+... +alnxn, x2 = a21x1+a22x2+... +a2nxn, x,n = a,nl xl + anal x2 + ... + a,nn xn, aif a F, and the coefficients aif determine the homomorphism t uniquely with respect to the'chosen bases. The rectangular array of coefficients all a12 a21 a22 ... aln a2n ... amn is said to form a matrix, A, having m rows and n columns. aml am2 Note. , n} -* F. (Compare the abstract definition of a sequence (p.

Consider the set S° c V* of all linear forms f on V which satisfy f(a) = o for all a e S. The set S° is a subspace of V*, called the annihilator of S in V*, and the spaces S and S° are said to be orthogonal. Note. It follows that dim S + dim (annihilator S) = dim V. Vector spaces and matrices 45 R will be of the Example. , xn) H allxl+... , n. , n (see p. , aln). The set of all such n-tuples (each one of which represents a welldefined linear functional) forms a vector space V*, the dual space of V, which is isomorphic to V.