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When E = F and the basis (v1 , . . , vm ) is identical to the basis (u1 , . . , un ) of E, the matrix M (f ) associated with f : E → E (as above) is called the matrix of f with respect to the base (u1 , . . , un ). 12, there is no reason to assume that the vectors in the bases (u1 , . . , un ) and (v1 , . . , vm ) are ordered in any particular way. However, it is often convenient to assume the natural ordering. When this is so, authors sometimes refer to the matrix M (f ) as the matrix of f with respect to the ordered bases (u 1 , .

3 CHAPTER 3. DETERMINANTS Definition of a Determinant Recall that the set of all square n × n-matrices with coefficients in a field K is denoted by Mn (K). , a map of the n columns of a matrix, is n-linear alternating and such that D(In ) = 1 for the identity matrix In . Equivalently, we can consider a vector space E of dimension n, some fixed basis (e1 , . .

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 . Polynomials over a commutative ring also form a free module of infinite dimension. Square matrices provide a natural example of a noncommutative ring with zero divisors. 12 For example, letting A, B be the 2 × 2-matrices A= 1 0 0 0 , 0 0 1 0 B= , then AB = 1 0 0 0 0 0 1 0 = 0 0 0 0 , BA = 0 0 1 0 1 0 0 0 = 0 0 1 0 . and We now formalize the representation of linear maps by matrices. 7. 14 Let E and F be two vector spaces, and let (u1 , .

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Basics of Algebra and Analysis for Computer Science by Gallier J.

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