By Gallier J.

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Un ). We have M (f ) = P −1 M (f )P. 8 Direct Sums Before considering linear forms and hyperplanes, we define the notion of direct sum and prove some simple proositions. There is a subtle point, which is that if we attempt to define the direct sum E ⊕ F of two vector spaces using the cartesian product E × F , we don’t quite get the right notion because elements of E × F are ordered pairs, but we want E ⊕ F = F ⊕ E. Thus, we want to think of the elements of E ⊕ F as unordrered pairs of elements.

22, we have the following result. 23 Let E and F be vector spaces, and let f : E → F be a linear map. Then, E is isomorphic to Ker f ⊕ Im f , and thus, dim(E) = dim(Ker f ) + dim(Im f ). Proof . Consider i i f Ker f −→ E −→ Im f, f where Ker f −→ E is the inclusion map, and E −→ Im f is the surjection associated f s with E −→ F . 19, to get dim(E) = dim(Ker f ) + dim(Im f ). The following Proposition will also be useful. 24 Let E be a vector space. If E = U ⊕ V and E = U ⊕ W , then there is an isomorphism f : V → W between V and W .

Ih } and {j1 , . . , jq , . . jk }, where ip = i and jq = j, since τ (π(π −1 (ip ))) = τ (ip ) = τ (i) = j = jq and τ (π(π −1 (jq ))) = τ (jq ) = τ (j) = i = ip , we see that the classes Jl and Jm merge into a single class, and thus, the number of classes associated with τ ◦ π is r − 1, and (τ ◦ π) = (−1)n−r+1 = −(−1)n−r = − (π). Now, let π = τm ◦ . . ◦ τ1 be any product of transpositions. By the first part of the proposition, we have (π) = (−1)m−1 (τ1 ) = (−1)m−1 (−1) = (−1)m , since (τ1 ) = −1 for a transposition.

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