By Muhammer Taşkıran, Cünet Kılıç

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We shall establish a fundamental relation concerning the nullity of F. T H E O R E M 26 Let X and Y be vector spaces over a field L, and let F be a linear transformation from X into Y. Then dim N(F) = dim X - dim tf(F). (23) PROOF Assume dim X = η and dim N(F) — η — k. Since N(F) (Ζ X, SL 1 k+1 k+2 7 basis {x , x , . , x *} of N(F) can be extended to a basis {JC , . , 1 k+1 w k JC*, x , . , x } of A". The set { ^ ( Λ : ) , . . , F(x )} is found to be linearly independent since if the set were linearly dependent, we would have θ= hatF(j<*) F[iiai*) I for some not-all-zero scalars ai^L (/ = 1, .

Assume η ^ 2. If \A\ = 0, then r k ( ^ ) ^ η — 1. Hence, by taking (24) into account, we know dim7V(,4) 1, which implies the existence of a nonzero vector χ satisfying Ax = 0. If | Λ | ^=0, then dim N(A) = 0, implying that the only solution to Ax = 0 is a null vector. D. Next we shall study a nonhomogeneous system of linear equations (25) where A is an m χ η matrix, d is a nonzero column w-vector, and χ is an unknown w-vector to be determined. If vk(A) = m, then (25) has a solution * for any d since rk[y4|i/] = m.

Hence, χ has a unique representation as the sum of unique linear combinations of these u's and w's, which implies that {#1, . , vp, w\, . , wq] constitutes a basis of X, proving the assertion. D. 2 Linear Transformations D E F I N I T I O N 17 Let X, y be arbitrary sets and D a subset in X. A rule F that associates an element y G Y to each element χ of D is said to be a iriwjformation (or mapping) from X into (or to) 7 with domain Z), and we write y = F(x) for x œ D œ or F : X-> Y. X and y œ Y9 40 LINEAR EQUATIONS W I T H REFERENCE T O ECONOMICS 2 If a domain is n o t explicitly specified with respect to a transformation defined on a set X, it should be understood that the domain is A" itself.

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