By Peter Wolff

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Am1 am2 · · · amn xn bm and define the product Ax by ⎧ a11 x1 + a12 x2 + · · · + a1n xn ⎪ ⎪ ⎪ ⎪ ⎪ a ⎪ 21 x1 + a22 x2 + · · · + a2n xn ⎪ Ax = ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ am1 x1 + am2 x2 + · · · + amn xn ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (5) then the linear system of equations (3) is equivalent to the matrix equation (4). Given an m × n matrix A and a vector x in Rn , it is possible to compute a product Ax by (5). The product Ax will be an m × 1 matrix, that is, a vector in Rm . The rule for determining the ith entry of Ax is ai1 x1 + ai2 x2 + · · · + ain xn which is equal to ai x, the scalar product of the ith row vector of A and the column vector x.

However, in Chapter 6 there will be occasions when we will use the set of complex numbers as our scalar field. Matrix Notation If we wish to refer to matrices without specifically writing out all their entries, we will use capital letters A, B, C, and so on. In general, aij will denote the entry of the matrix A that is in the ith row and the jth column. We will refer to this entry as the (i, j) entry of A. Thus, if A is an m × n matrix, then ⎫ ⎧ a11 a12 · · · a1n ⎪ ⎪ ⎪ ⎪ ⎪ a21 a22 · · · a2n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A=⎪ ⎪ ⎪ .

Thus the total value of goods received by the farmer is 12 x1 + 13 x2 + 12 x3 . If the system is fair, the total value of goods received by the farmers should equal x1 , the total value of the farm goods produced. 2 Row Echelon Form 39 Finally, using the third row of the table, we get 1 1 1 x1 + x2 + x3 = x3 4 3 4 These equations can be rewritten as a homogeneous system: − 12 x1 + 13 x2 + 12 x3 = 0 1 x − 23 x2 + 14 x3 = 0 4 1 1 x + 13 x2 − 34 x3 = 0 4 1 The reduced row echelon form of the augmented matrix for this system is ⎫ ⎧ 1 0 − 53 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 −1 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 There is one free variable: x3 .