By Gallier J.

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**Example text**

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 .