ON LINEAR ALGEBRA: VECTOR AND MATRIX CALCULUS 507 as a + µ 1 v 1 + · · · + µ k v k for a unique selection of numbers µ 1 , . . . , µ k , then we call k 0 the dimension of the solution set of the system. This is the professional concept for the intuitive concept "number of free variables in the general solution." It can be shown that for every solvable system such vectors a, v 1 , . . . , v k can be chosen and that for each choice we have that k, the number of v's, is the same. For each matrix A, we can choose a collection of rows such that each row of the matrix can be expressed as a linear combination of the rows of this collection in a unique way. The number of rows in this collection is called the rank of the matrix A. It can be shown that this number does not depend on the choice of the collection. The rank of the coefficient matrix of a given system is the professional concept for the intuitive concept "true number of equations." Thus the rule above can be given its precise formulation: the dimension of the solution set equals the number of variables minus the rank of the extended matrix, provided the system has any solutions at all.