not fewer. The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix. (This is the rank–nullity theorem.) If is a matrix over the real numbers then the rank of and the rank of its corresponding Gram matrix are equal. Thus, for real matrices This can be shown by proving equality of their null spaces. The null space of the Gram matrix is given by vectors for which If this condition is fulfilled, we also have If is a matrix over the complex numbers and denotes the complex conjugate of and the conjugate transpose of (i.e., the adjoint of ), then Applications One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. According to the Rouché–Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If on the other hand, the ranks of these two matrices are equal, then the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has free parameters where is the difference between the number of variables and the rank. In this case (and assuming the system of equations is in the real or complex numbers) the system of equations has infinitely many solutions. In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable. In the field of communication complexity, the rank of the communication matrix of a function gives bounds on the amount of communication needed for two parties to compute the function. Generalization There are different generalizations of the concept of rank to matrices over arbitrary rings, where column rank, row rank, dimension of column space, and dimension of row space of a matrix may be different from the others or may not exist. Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors; for tensors of order greater than 2 (matrices are order 2 tensors), rank is very hard to compute, unlike for matrices. There is a notion of rank for smooth maps between smooth manifolds. It is equal to the linear rank of the derivative. Matrices as tensors Matrix rank should not be confused with tensor order, which