By Alessandrini L.

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**Example text**

Consider the vector space, F, generated by the vectors x1 Xr+i Then by induction, F has a basis of the form X1 Xr, where V1ES Ys 1 (j = 1 Now consider the vector x,1. If the vectors x1 ). 1 x, are linearly independent, then they form a basis of E which has the desired property. Otherwise there is a non-trivial relation = 0. + Since the vectors x1 are that y+O. Thus linearly independent, it follows r x,7= Q=l 71 generate E. Since they are linearly Hence the vectors x1 independent, they form a basis.

5. Exact sequences. Exact sequences provide a sophisticated method for describing elementary properties of linear mappings. 7) Chapter II. Linear mappings 46 is called exact at E if Imp = We exhibit the following special cases: 1. F=O. , çti is injective. Conversely, is injective. 8) suppose Ji: is exact at E. 2. G=0. 9) Since ,fi is the zero mapping it follows that Imq = ker/i = E and so p is surjective. 9) is exact. 10) which is exact at F, E and G. 11) where E1 is a subspace of E and 1, projection respectively.

K/Jo p will often be denoted simply by k/1p. If p is a linear transformation in E, then we denote (/9 o (p by (p2. More generally, the linear transformation p ... p is denoted by shows that k/Jo i. A linear transformation, p, satisfying (p2 = i is called an involution in E. 11. Generators and basis. S—*F Proposition I: Suppose S is a system of generators for E and is a set map (Fa second vector space). Then Po can be extended in at most one way to a linear mapping We extend the definition to the case k=O by setting (p°= F.