functions in whose derivatives are zero, i.e. the set of all constant functions. Let be the direct product of infinitely many copies of , and let be the shift operator Then the kernel of s is the one-dimensional subspace consisting of all vectors . If is an inner product space and is a subspace, the kernel of the orthogonal projection is the orthogonal complement to in . Computation by Gaussian elimination A basis of the kernel of a matrix may be computed by Gaussian elimination. For this purpose, given an m × n matrix A, we construct first the row augmented matrix where is the n × n identity matrix. Computing its column echelon form by Gaussian elimination (or any other suitable method), we get a matrix A basis of the kernel of A consists in the non-zero columns of C such that the corresponding column of B is a zero column. In fact, the computation may be stopped as soon as the upper matrix is in column echelon form: the remainder of the computation consists in changing the basis of the vector space generated by the columns whose upper part is zero. For example, suppose that Then Putting the upper part in column echelon form by column operations on the whole matrix gives The last three columns of B are zero columns. Therefore, the three last vectors of C, are a basis of the kernel of A. Proof that the method computes the kernel: Since column operations correspond to post-multiplication by invertible matrices, the fact that reduces to means that there exists an invertible matrix such that with in column echelon form. Thus and A column vector belongs to the kernel of (that is ) if and only if where As is in column echelon form, if and only if the nonzero entries of correspond to the zero columns of By multiplying by , one may deduce that this is the case if and only if is a linear combination of the corresponding columns of Numerical computation The problem of computing the kernel on a computer depends on the nature of the coefficients. Exact coefficients If the coefficients of the matrix are exactly given numbers, the column echelon form of the matrix may be computed by Bareiss algorithm more efficiently than with Gaussian elimination. It is even more efficient to use modular arithmetic and Chinese remainder theorem, which reduces