make two observations: (a) is a linear combination of vectors in the row space of , which implies that belongs to the row space of , and (b) since , the vector is orthogonal to every row vector of and, hence, is orthogonal to every vector in the row space of . The facts (a) and (b) together imply that is orthogonal to itself, which proves that or, by the definition of , But recall that the were chosen as a basis of the row space of and so are linearly independent. This implies that . It follows that are linearly independent. Now, each is obviously a vector in the column space of . So, is a set of linearly independent vectors in the column space of and, hence, the dimension of the column space of (i.e., the column rank of ) must be at least as big as . This proves that row rank of is no larger than the column rank of . Now apply this result to the transpose of to get the reverse inequality and conclude as in the previous proof. Alternative definitions In all the definitions in this section, the matrix is taken to be an matrix over an arbitrary field . Dimension of image Given the matrix , there is an associated linear mapping defined by The rank of is the dimension of the image of . This definition has the advantage that it can be applied to any linear map without need for a specific matrix. Rank in terms of nullity Given the same linear mapping as above, the rank is minus the dimension of the kernel of . The rank–nullity theorem states that this definition is equivalent to the preceding one. Column rank – dimension of column space The rank of is the maximal number of linearly independent columns of ; this is the dimension of the column space of (the column space being the subspace of generated by the columns of , which is in fact just the image of the linear map associated to ). Row rank – dimension of row space The rank of is the maximal number of linearly independent rows of ; this is the dimension of the row space of . Decomposition rank The rank of is the smallest integer such that can be factored as , where is an matrix and is a matrix. In