An Introduction to Homological Algebra by Charles A. Weibel

By Charles A. Weibel

A portrait of the topic of homological algebra because it exists this day

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5. The following example is somewhat more difficult. 21 Many questions in physics (such as the distribution of heat in a solid body if a known temperature is maintained on its surface, or the distribution of electric charge on a body if a known charge distribution is maintained on its surface, and so on) lead to a single differential equation, called the Laplace equation. It is a partial differential equation, which we do not need to describe here. It suffices to mention one consequence, called the mean value property, according to which the value of the unknown quantity (satisfying the Laplace equation) is equal at every point to the arithmetic mean of its values at “nearby” points.

7) is not equal to zero. 7) again gives a solution, namely the null solution, but now we can no longer assert that all solutions are obtained by multiplying by a number (indeed, this is not true). Let us now consider the case of a system of three equations in three unknowns: ⎧ ⎪ ⎨a11 x1 + a12 x2 + a13 x3 = b1 , a21 x1 + a22 x2 + a23 x3 = b2 , ⎪ ⎩ a31 x1 + a32 x2 + a33 x3 = b3 . We again would like to eliminate x2 and x3 from the system in order to obtain a value for x1 . To this end, we multiply the first equation by c1 , the second by c2 , 28 2 Matrices and Determinants and the third by c3 and add them.

N (see the definition of linear function on p. 2). We begin with the term ±ai1 Mi1 . Since the minor Mi1 does not depend on the elements of the ith row—the ith row is ignored in the calculation—it is simply a constant as a function of the ith row. Let us set d1i = ±Mi1 and d2i = d3i = · · · = dni = 0. Then the first term is represented in the required form, and indeed is a linear function of the ith row of the matrix A. For the term ±aj 1 Mj 1 , for j = i, the element aj 1 does not appear in the ith row, but all the elements of the ith row of matrix A other than ai1 appear in some row of the minor Mj 1 .

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