1.2. Vecto r spaces 7 Recall (or see section A.2 in the appendix) that a map p: X i— • F from a set X to a set F is bijective if for every y eY ther e is precisely one xeX suc h that y = (p(x) . Bijective maps are invertible—the inverse map is defined by: (1.2.1) (p~ l (y)=x i f y = p(x). Let y an d W b e vector spaces (over the same field). DEFINITION: A map p: ^ i- W i s //near if for all scalars (3, & and vectors vl5v2 G ^ (1.2.2) cp(av 1 +fev2) =ajo(v 1 ) + fe(jp(v 2 ). A map p: Y \-+ W is an isomorphism if it is botf? bijective and linear. An automorphism is an isomorphism of a vector space y ont o itself. If (p is an isomorphism of y ont o #^, then (p~ l is an isomorphism of W ont o y. Thi s can be seen as follows: by (1.2.2), 2 3 ) P~1(ai(Pvi + a 2^ v 2) = P"1((P(^iV 1 +a2v2)) = a l vl+a2v2 = a l (p-1((pvl)+a2(p-1(qv2), and, a s p is surjective , ever y vecto r i n W i s equa l t o (pv for som e We say that y and W are isomorphic if there is an isomorphism cp of y ont o W, an d the fact that the inverse (p~l is also an isomorphism guarantees that the relation of being isomorphic is symmetric. The identity map (p{x) — x shows that the relation is reflexive and, since the composition o f isomorphisms i s an isomorphism, se e exer- cise exl.2.2, the relation is also transitive. In other words, the relations of being isomorphic is an equivalence relation (see A. 1.2). 1.2.3 Subspaces . A (vector) subspace of a vector space y i s a sub- set that is closed under the operations o f additio n an d multiplicatio n by scalars inherited from y. I n other words, W C y i s a subspace if for all scalars a- and vectors w G W, j = 1,2 , the vectors axwx + a 2 w2 are in W.

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