# Download e-book for kindle: Seminaire Bourbaki 1972-1973, exposes 418-435 by Bourbaki N.

By Bourbaki N.

ISBN-10: 0387067965

ISBN-13: 9780387067964

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Extra info for Seminaire Bourbaki 1972-1973, exposes 418-435

Sample text

We can also use this to study maps V ! W as long as we have bases e1 ; : : : ; em for V and f1 ; : : : ; fn for W . Each x 2 V has a unique expansion x D e1 So if L W V ! ei /. e1 / as linear maps. em W Fm ! em / W Fm ! ei / D xi with respect to the basis for W gives us xi D f1 fn ˇni and x1 3 ˇ1i 6 :: 7 4 : 5 2 xm D f1 fn 2 3 ˇ11 ˇ1m 6 :: : : :: 7 4 : : : 5: ˇnm ˇn1 This gives us the matrix representation for a linear map V ! W with respect to the specified bases. x/ D x1 D f1 xm 2 fn 2 3 ˛1 6 :: 7 4 : 5 ˛m 32 3 ˛1 ˇ11 ˇ1m 6 :: : : :: 7 6 :: 7 4 : : : 54 : 5: ˇnm ˇn1 ˛m We will often use the notation 2 3 ˇ11 ˇ1m 6 7 ŒL D 4 ::: : : : ::: 5 ˇnm ˇn1 for the matrix representing L.

F/ is the matrix whose j th column is the i th column of A and all other entries are zero. 9. Let e1 ; e2 be the standard basis for C2 and consider the two real bases e1 ; e2 ; i e1 ; i e2 and e1 ; i e1 ; e2 ; i e2 . If D ˛ C iˇ is a complex number, then compute the real matrix representations for 1C2 with respect to both bases. 10. Show that if L W V ! V has a lower triangular representation with respect to the basis x1 ; : : : ; xn ; then it has an upper triangular representation with respect to xn ; : : : ; x1 .

2. Two vector spaces V and W over F are said to be isomorphic if we can find an isomorphism L W V ! W . Note that if V1 and V2 are isomorphic and V2 and V3 are isomorphic, then V1 and V3 are also isomorphic. The isomorphism is the composition of the given isomorphisms. Recall that a map f W S ! x2 / implies that x1 D x2 . A better name for this concept is two-to-two as pointed out by Arens, since injective maps evidently take two distinct points to two distinct points. We say that f W S ! x/ for some x 2 S .