By Philip S. Hirschhorn

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**Example text**

Then f is a linear form on V. l. If T:V- W is a linear mapping, then T(x- y) = Tx- Ty for all x, y in V. 4. If V is a vector space, prove that the mapping T:V x V- V defined by T(x, y) = x- y is linear. ) 5. 3, an element (a1o .. , an) of Fn such that T(ah . • , Xn. For example, suppose that V = F 2, n = 3 and x 1 = (2, -3), x 2 = (4, 1), x 3 = (8, 9). 3. (ii) Show that ( -2, 3, -1) is a relation among xh x 2, x 3 • 6. 2 uses only the additivity of the mapping T. Give a proof using only its homogeneity.

The statement [x] C X may be written [x] e "P(X), where tp(X) is the power set 2 of X; thus X/- is a subset of tp(X), that is, X/- c tp(X). 4), then V/- is replaced by the more suggestive notation V/M. 8). 17 Definition If - is an equivalence relation in a set X, there is a natural mapping q:X-+ X/-, namely, the mapping that assigns to each point x in X its equivalence class [x]: for all x eX. 7. 5 EQUIVALENCE RELATIONS AND QUOTIENT SETS 45 Since X/- is the set of all classes [x ], it is obvious that q is surjective; it is called the quotient mapping of X onto X/-.

11} and y is a fixed vector in W, the mapping T:V-+ W defined by Tx = l(x)y for all x e V is linear. } The dependence of T on I and y is expressed by writing T =I® y (which suggests a kind of 'product' of I and y). (V, W}, (I® y)(x) = l(x)y for all x e V. (V); T is called a scalar linear mapping. (F, V). 11). (V, W). Can it be made into a vector space? Why should we ask the question? At any rate, the problem would be to define sums and scalar multiples of linear mappings, and the motivation should come from 'real life'.