By Gallier J.
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This variation reproduces the second corrected printing of the 3rd variation of the now vintage notes via Professor Serre, lengthy validated as one of many regular introductory texts on neighborhood algebra. Referring for historical past notions to Bourbaki's "Commutative Algebra" (English variation Springer-Verlag 1988), the e-book focusses at the quite a few size theories and theorems on mulitplicities of intersections with the Cartan-Eilenberg functor Tor because the valuable inspiration.
New version comprises broad revisions of the cloth on finite teams and Galois thought. New difficulties extra all through.
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We take a quick look at the connection between E and L(E; K), its dual space. 21 Given a vector space E, the vector space L(E; K) of linear maps f : E → K is called the dual space (or dual) of E. The space L(E; K) is also denoted by E ∗ , and the linear maps in E ∗ are called the linear forms, or covectors. The dual space E ∗∗ of the space E ∗ is called the bidual of E. As a matter of notation, linear forms f : E → K will also be denoted by starred symbol, such as u∗ , x∗ , etc. Given a linear form u∗ ∈ E ∗ and a vector v ∈ E, the result u∗ (v) of applying u∗ to v is also denoted by u∗ , v .
Am 1 am 2 . . a m n bn 1 bn 2 ... ... b1 p b2 p = .. . . . bn p c1 p c2 p .. . c1 1 c2 1 .. c1 2 c2 2 .. ... ... cm 1 cm 2 . . cm p note that the entry of index i and j of the matrix AB is obtained by multiplying the matrices A and B can be identified with the product of the row matrix corresponding to the i-th row of A with the column matrix corresponding to the j-column of B: b1 j n ai k b k j ) (ai 1 , . . , ai n ) ... = ( k=1 bn j The square matrix In of dimension n containing 1 on the diagonal and 0 everywhere else is called the identity matrix .
The theory of modules is (much) more complicated than that of vector spaces. For example, modules do not always have a basis, and other properties holding for vector spaces usually fail for modules. When a module has a basis, it is called a free module. For example, when A is a commutative ring, the structure An is a module such that the vectors ei , with (ei )i = 1 and (ei )j = 0 for j = i, form a basis of An . Many properties of vector spaces still hold for An . Thus, An is a free module. As another example, when A is a commutative ring, Mm,n (A) is a free module with basis (Ei,j )1≤i≤m,1≤j≤n .