BOUNDARY PROBLEMS AND DEVELOPMENTS. 65 



The number of linearly independent solutions of the linear algebraic 

 system (9), (and hence of equation (8)) is, however, precisely equal to 

 the difference between the number of equations, n, and the rank of the 

 determinant | pa |. Thus when | W a Y(a) + W b Y(b) | = 0, and is 

 of rank (n — k), then there are precisely k linearly independent solu- 

 tions Cy , Ci',. . .Ch', of equation (8), and correspondingly just k 

 linearly independent solutions Yi(x)- = Y(x) d- i= l,2,...k, 

 of system (6). Conversely if there are k independent solutions of 

 system (6) the rank of the determinant is (n — k). 



Suppose then that system (0) is known to have just k linearly inde- 

 pendent solutions. This means that the determinant vanishes and is 

 of rank (n—k), and since | XiU X 2 | is of the same rank as | U | when 

 | A r x | =J= 0, and | X 2 | 4= 0, 5 it follows that 



| Z(a) Wa-' { W a Y(a) + W h Y(b) } Z(b) W b ~ l | = 



and is of rank (n — k), Z{x) being any matrix solution of equation 

 (2). If in particular Zix) is chosen as the solution associated with 

 Y(x) this reduces to the statement that | Zifl) W a ~ l + Z{b) W b ~ l \ = 0, 

 and is of rank (n — k), which implies that system (7) also has just 

 k linearly independent solutions. Thus the theorem is proved. 



A system of the type (6) or (7) is said to be either compatible or in- 

 compatible according as it does or does not admit of a solution not 

 identically 0. Its compatibility is said to be A: -fold when the number 

 of its linearly independent solutions is k. 



Consider the non-homogeneous system 



I Y'(x)- =A(x)Y(x)-+B(x)- 

 K } \ W a Y(a)- +W h Y(b)- =0. 



Theorem: A necessary and sufficient condition that system (10) 

 has a unique solution is that the corresponding homogeneous system 

 (6) is incompatible. 



Proof: It was shown (formula (5)) that the general solution of equa- 

 tion (3) is 



Y(x) ss Y h (x) C + Y(x) 

 5 Cf. Bocher, loc. cit., pp. 77-79. 



