Mathematical Methods for Physics and Engineering - Matematica.NET

8.18 SIMULTANEOUS LINEAR EQUATIONSwhere the A ij and b i have known values. If all the b i are zero then the system ofequations is called homogeneous, otherwise it is inhomogeneous. Depending on thegiven values, this set of equations for the N unknowns x 1 , x 2 , ..., x N may haveeither a unique solution, no solution or infinitely many solutions. Matrix analysismay be used to distinguish between the possibilities. The set of equations may beexpressed as a single matrix equation Ax = b, or, written out in full, as⎛ ⎞⎛⎞ ⎛ ⎞ bA 11 A 12 ... A 1N x 11A 21 A 22 ... A 2Nx 2b ⎜⎝. ⎟ ⎜ ⎟. . .. . ⎠ ⎝ . ⎠ = 2.⎜⎝ . ⎟⎠A M1 A M2 ... A MNx Nb M8.18.1 The range and null space of a matrixAs we discussed in section 8.2, we may interpret the matrix equation Ax = b asrepresenting, in some basis, the linear transformation A x = b of a vector x in anN-dimensional vector space V into a vector b in some other (in general different)M-dimensional vector space W .In general the operator A will map any vector in V into some particularsubspace of W , which may be the entire space. This subspace is called the rangeof A (or A) and its dimension is equal to the rank of A. Moreover, if A (andhence A) issingular then there exists some subspace of V that is mapped ontothe zero vector 0 in W ; that is, any vector y that lies in the subspace satisfiesA y = 0. This subspace is called the null space of A and the dimension of thisnull space is called the nullity of A. We note that the matrix A must be singularif M ≠ N and may be singular even if M = N.The dimensions of the range and the null space of a matrix are related throughthe fundamental relationshiprank A + nullity A = N, (8.119)where N is the number of original unknowns x 1 ,x 2 ,...,x N .◮Prove the relationship (8.119).As discussed in section 8.11, if the columns of an M × N matrix A are interpreted as thecomponents, in a given basis, of N (M-component) vectors v 1 , v 2 ,...,v N then rank A isequal to the number of linearly independent vectors in this set (this number is also equalto the dimension of the vector space spanned by these vectors). Writing (8.118) in termsof the vectors v 1 , v 2 ,...,v N , we havex 1 v 1 + x 2 v 2 + ···+ x N v N = b. (8.120)From this expression, we immediately deduce that the range of A is merely the span ofthe vectors v 1 , v 2 ,...,v N and hence has dimension r =rankA.293