Advanced Calculus fi..

Chapter 1 Vectors and Matrices 65the unknowns in terms of the remaining unknowns. By renumbering we can assumethat xl, . . . , x, are expressed in terms of sr+l. . . , x,,:Here we writeThen the solutions are given bywhere w,+~, . .. , w, are arbitrary scalars. Thus the kernel of A is represented as theset of linear combinations of n - r vectors u,+~, .... u,,. We see that these vectorsare linearly independent (see Problem 5 below). Therefore the kernel has dimensionn - r or h = n - r. Equation (1.124) is proved. [Note: As in Section 1.10, if r = 0,Ax = 0 is satisfied by all x, so that h = n.]Maximum rank. Determinant deJinition of rank. For a k x n matrix A we haveseen that the rank r of A cannot exceed k or n. When r is the largest integer satisfyingthis condition, A is said to have tnaximurn rank. If k 5 n, this means that r = k; ifk > n, it means that r = n.By a minor of A we mean a determinant formed from the array A by strikingout certain rows and columns to obtain a square array. If, for example, A is 3 x 3,then A has one 3 x 3 minor det A, nine 2 x 2 minors, and nine 1 x I minors (thenine entries of A).One can show that r, the rank of A, is the largest integer such that some r x rminor of A is nonzero (see Problem 6, which follows).EXAMPLE 3 We again consider the matrix A of Example 1. The kernel of A isfound from the Gaussian elimination applied to A, as was done in Example 1. Henceit consists of all x such that- 10-x2 + 3x3 - X', = 0.We solve for x,, x2, letting x3 = w3, x4 = w4. We find