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XVI DIFFERENTIAL MANIFOLDS

form a frame of F satisfying condition (2) of (16.17.1) relative to I. On the
other hand, Cramer's formulas show that

where the ()_ki are of class C°°; and the fact that N is a vector subbundle of E
follows because the sections

p \

form a frame of E satisfying condition (2) of (16.17.1) relative to N.

When the conditions of (16.17.5(ii)) are satisfied, the vector bundles N
and I are called, respectively, the kernel and the image of u and are written
Ker(tt) and Im(w). If p : E ~> E/Ker(w) and j: Im(w) -» F are the canonical
morphisms, then the unique mapping v : E/Ker(w) -> Im(w) such that u =
j ° v op is an isomorphism of vector bundles (16.15.2).

(16.17.6) Let E A F -^ G be a sequence of two morphisms of vector bundles
over B. The sequence is said to be exact if, for each b e B, the sequence

_ft

of linear mappings is exact. A finite sequence

of morphisms of vector bundles over B is said to be exact if each of the

sequences E_k ~~"*** >~~ E_k+i ~~^Uk+2 >~~ E_k+2(Q ^k<>n -2) is exact.

If we denote by 0 the trivial bundle B x {0}, then a morphism u : E -*• F
of vector bundles over B is infective (resp. surjective) if and only if the sequence
0 — * E A F (resp. E A F -> 0) is exact. For each vector subbundle E' of E, the
sequence

(with the notation of (16.17.1) and (16.17.2)) is exact.

(1 6.17.7) IfE --* F -^ G is an exact sequence of morphisms of vector bundles
over B, then u and v satisfy the equivalent conditions of (16.17.5(ii)), and the
bundles Im(w) and Ker(y) are therefore defined and equal.



	XVI DIFFERENTIAL MANIFOLDS form a frame of F satisfying condition (2) of (16.17.1) relative to I. On the other hand, Cramer's formulas show that where the ()_ki are of class C°°; and the fact that N is a vector subbundle of E follows because the sections p \

	form a frame of E satisfying condition (2) of (16.17.1) relative to N. When the conditions of (16.17.5(ii)) are satisfied, the vector bundles N and I are called, respectively, the kernel and the image of u and are written Ker(tt) and Im(w). If p : E ~> E/Ker(w) and j: Im(w) -» F are the canonical morphisms, then the unique mapping v : E/Ker(w) -> Im(w) such that u = j ° v op is an isomorphism of vector bundles (16.15.2). (16.17.6) Let E A F -^ G be a sequence of two morphisms of vector bundles over B. The sequence is said to be exact if, for each b e B, the sequence

	_ft of linear mappings is exact. A finite sequence

	of morphisms of vector bundles over B is said to be exact if each of the sequences E_k ~~"*** >~~ E_k+i ~~^Uk+2 >~~ E_k+2(Q ^k<>n -2) is exact. If we denote by 0 the trivial bundle B x {0}, then a morphism u : E -• F of vector bundles over B is infective* (resp. surjective) if and only if the sequence 0 — * E A F (resp. E A F -> 0) is exact. For each vector subbundle E' of E, the sequence

	(with the notation of (16.17.1) and (16.17.2)) is exact. (1 6.17.7) IfE --* F -^ G is an exact sequence of morphisms of vector bundles over B, then u and v satisfy the equivalent conditions of (16.17.5(ii)), and the bundles Im(w) and Ker(y) are therefore defined and equal.