By Marcel Dekker, Inc

Goals and Scope. This magazine provides full-length articles thai replicate major advances in all components of present algebraic curiosity and task.

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22 Linear Equations Chap. 1 m X m identity matrix. In other words, there is an m X m matrix Q, which is itself a product of elementary matrices. such that QP = I. As we shall soon see, the existence of a Q with QP = I is equivalent to the fact that P is a product of elementary matrices. DeJinition. Let A be an n X n (square) matrix over the field F. An n X n matrix B such that BA = I is called a left inverse of A; an n X n matrix B such that AB = I is called a right inverse of A. If AB = BA = I, then B is called a two-sided inverse of A and A is said to be invertible.

The n-tuple space, F n. Let F be any field, and let V be the set of all n-tuples (Y = (x1, Q, . . , 2,) of scalars zi in F. If p = (Yl, Yz, . . , yn) with yi in F, the sum of (Y and p is defined by (2-l) The product (2-2) a + P = (21 + y/1, 22 of a scalar c and vector + yz, . f f , & + Y/n>. LYis defined by ca = (CZl, cz2, . . , CZJ . The fact that this vector addition and scalar multiplication satisfy conditions (3) and (4) is easy to verify, using the similar properties of addition and multiplication of elements of F.

Y,,,in S is clearly in W. Thus W contains the set L of all linear combinations of vectors in S. The set L, on the other hand, contains S and is non-empty. If (Y, /3 belong to L then CYis a linear combination, CY= Xlffl + x2ayz+ * f * + XmQ, of vectors (pi in S, and ,B is a linear combination, P = of vectors YlPl + Y2P2 + * * * + Y&I @j in S. For each scalar c, Cff + P = 5 (CXi)ai +jgl yjPj* i=l Hence ca! + ,Obelongs to L. Thus L is a subspace of V. Now we have shown that L is a subspace of V which contains S, and also that any subspace which contains S contains L.

### Communications in Algebra, volume 25, number 9, 1997 by Marcel Dekker, Inc

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