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This follows from the distributivity of matrix multiplication over addition. We also touch on the row space and its relation to the kernel. A are orthogonal to each of the row vectors of A. 3, we have an illustration of the rank-nullity theorem. 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.