13. Systems of Linear Equations 1

Transcription

1 13. Systems of Linear Equations 1 Systems of linear equations One of the primary goals of a first course in linear algebra is to impress upon the student how powerful matrix methods are in solving systems of linear equations (the single most important application of linear algebra). We will now summarize those ideas. The system a 11 + a 12 +L + a 1n = b 1 a 21 + a 22 +L + a 2n = b 2 a m1 + a m 2 +L + a mn = b m of m linear equations in n variables has a muchabbreviated matrix form where AX = B A = a 11 a 12 L a 1n a 21 a 22 L a 2n L a m1 a m 2 L a mn, X =, and B = b 1 b 2 b n.

2 13. Systems of Linear Equations 2 When B = 0 in R mn, we call the system homogeneous; otherwise, we say it is affine. There is an intimate relationship between the affine system AX = B and its homogeneous counterpart AX = 0. Proposition Let T: R n R m be the linear transformation represented in terms of the standard bases for the underlying vector spaces by the matrix A, i.e., T(X) E = AX F, where the ordered bases {E 1, E 2,, E m } and {F 1, F 2,, F n } are the standard bases for R m and R n, repsectively. Then (1) the solution set of the homogeneous system AX = 0 is a subspace of R n, namely, ker(t). Furthermore, either (2) the solution set U of the affine system AX = B is empty when B Im(T ), or, (3) when B Im(T ), the solution set U of the affine system AX = B is a parallel translate of ker(t), that is, there is a fixed vector U R n such that U = U + ker(t) = {U + X AX = 0}. Finally, (4) if ker(t) is nontrivial, then U is not uniquely represented by the particular solution X = U: there are other U U that satisfy AX = B, and U = U + ker(t) as well. //

3 13. Systems of Linear Equations 3 The simplex method Recall that solving a system of linear equations a 11 + a 12 +L + a 1n = b 1 a 21 + a 22 +L + a 2n = b 2 a m1 + a m 2 +L + a mn = b m can be carried out efficiently by bringing it to reduced row echelon form: represent the system in the form of a matrix equation AX = B with A = a 11 a 12 L a 1n a 21 a 22 L a 2n L a m1 a m 2 L a mn, X =, and B = b 1 b 2 b n ; then, forming the augmented matrix ( A B) = a 11 L a 1n b 1 a 21 L a 2n b 2, L a m1 L a mn b m we transform it into another augmented matrix

4 13. Systems of Linear Equations 4 ( A B ) = a 11 L a 1n b 1 a 21 L a 2n b 2, L a m1 L a mn b m which represents the matrix equation A X = B. This transformation procedure does not change the solution set of the system, so the final equation has exactly the same solutions as the original. But since the transformed system is much easier to solve, we have gained much in the process. ore specifically, the augmented matrix ( A B ) is in reduced row echelon form when in every row, the first nonzero entry is a 1 and every other entry in the same column is a 0. For instance, is in reduced row echelon form; it represents the system

5 13. Systems of Linear Equations 5 + x 3 + x 5 = 0 + x 3 + 2x 5 = 2 x 4 + 3x 5 = 0 0 = 0 which is trivial to solve (from the bottom equation up, by back substitution). We find that (,,x 3,x 4, x 5 ) = ( x 3 x 5,2 x 3 2x 5, x 3, 3x 5,x 5 ) = (0,2,0,0,0)+ x 3 ( 1, 1,1,0,0)+ x 5 ( 1, 2,0, 3,1) provides a complete solution to the system. The transformation process that carries AX = B to A X = B is achieved by means of a sequence of elementary row operations of the following three types: I. interchange the ith equation with the jth equation; II. multiply through the ith equation by the nonzero scalar r; III. replace the ith equation by its sum with the jth equation. As none of these operations alter the conditions of any of the equations it affects, any sequence of them will not change the solution set of the system.

6 13. Systems of Linear Equations 6 Now, while this process is straightforward for use by human computers dealing with small numbers of equations and unknowns, there is another procedure that is slightly more difficult to describe but much faster than row echelon reduction when implemented on an electronic computer. This method, the simplex method, does essentially the same thing: it takes a given system of linear equations with matrix equation AX = B, and represents it by means of a matrix A B X tr 0 called a tableau. Note that if A is m n, then the tableau is (m +1) (n +1); further, the tableau records the variables of the system in its bottom row, each variable footing the column that contains its coefficients. This tableau is transformed into other tableaux by means of an operations called a pivot. In contrast to row reduction, there is only one kind of pivot, but it is complicated to state formally. Still, here it is. The ith row of the tableau represents the equation a i1 +a i2 +L+a in = b i. If 0, then we can solve the equation for x j to rewrite it in the form

7 13. Systems of Linear Equations 7 (*) a i1 a i2 L+ b i L a in = x j. Notice how this operation has replaced an equation which expresses the known value b i as a linear combination of the unknowns,,, with one that expresses the variable x j as a linear combination of the remaining unknowns, now together with the known b i. Substituting this expression for x j into the other equations of the system allows us to replace them with equations in which x j does not appear. For instance, equation k ( i) is transformed from into a k1 + a k2 +L+a kj x j +L+a kn = b k (**) a k1 a i1 a kj a + a k2 a i2 a kj ij a +L ij + a kj b j +L+ a kn a in a kj a ij a = b k ij For each of the values of k (except k = i) this replaces an equation which expresses the known

8 13. Systems of Linear Equations 8 value b k as a linear combination of the unknowns,,, with one that expresses b k as a linear combination of the unknowns, with b i replacing the unknown x i. If we now replace the original ith equation with (*) and all other equations with the corresponding version of (**), then following the pivot, the original tableau becomes the transformed tableau A B X tr 0 where X and B are identical with X and B except that x i in X has been swapped with b i from B; also, A = ( ) is the coefficient matrix obtained from A satisfying the formulas a rs = a rs a is 1 a is 1 a rj 1 a rj 1 r = i, s = j r = i, s j r i, s = j r i, s j where the notation denotes a 2 2 determinant.

9 13. Systems of Linear Equations 9 Notice that in the new tableau, there is one fewer variable in the bottom row than before. The aim is to continue to perform pivot operations until all the unknowns have been swapped out of the bottom row and into the final column. If this can be accomplished, the augmented matrix ( A B ) will then provide solutions for each of the variables (or may indicate an impossible equality, showing that there are no solutions). For examples of the use of pivot operations on simplex tableaux, see pp of the text.