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3-dimensional apply augmented matrix basis change becomes row calculation change of basis change the basis choose a basis codomain basis elements codomain basis vectors dependent Determine the matrix Differential Equations dimension elementary matrix elementary row operation Equation 4.3 example Exercise ff(a find the Hermite find the inverse Gauss elimination method gives Hermite normal form Hilbert matrix images Linear Algebra linear combination Linear Mathematics linear problem linear relations linear transformation m x n matrix matrix multiplication matrix of transition matrix Q matrix representing matrix to Hermite non-singular matrix non-zero element non-zero entry notation number of linearly o(ak o(Uk obtained old basis page N55 page N64 problem page N64 Proposition 5.2 Q~lA reading passage representing a linear respect to bases row 1 becomes row-sum check Section 3.2 set of vectors solution set solving specified standard basis transition matrix transpose unit matrix vector space vector with respect zero