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Then, as the matrix product is associative,
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Then and matrix
Then the trace of the identity matrix is the dimension of the space ; this leads to generalizations of dimension using trace.
Then, the problem of computing the function value can be rephrased as " zeroing-in " on the corresponding matrix entry.
Then the matrices would progress through the machine, where a special keying system on one end of the matrix, unique for each character, would allow the matrix to drop only into the correct storage slot, ready for future use.
Then, the inverse of this matrix is S, the lower triangular matrix of Stirling numbers of second kind.
Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.
Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:
Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.
Then, the individual normal equation matrices were combined and the resultant matrix solved to obtain the positions and the parameters.
Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f ( being the product of the eigenvalues of the Hessian ).
Then, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set
Then and product
Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.
Then, in the early 550s, two monks succeeded in smuggling eggs of silk worms from Central Asia back to Constantinople, and silk became an indigenous product.
Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule
Then the assignment extends uniquely to an algebra homomorphism by sending the monomial in the Clifford algebra to the product of matrices and extending linearly.
Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.
Then R (( G )) is the ring of formal power series on G ; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.
Then Edwin Binney, working with his wife, Alice Stead Binney, developed his own famous product line of wax crayons beginning on 10 June 1903, which it sold under the brand name " Crayola.
Then the unrestricted wreath product A Wr < sub > Ω </ sub > H of A by H is the semidirect product K ⋊ H. The subgroup K of A Wr < sub > Ω </ sub > H is called the base of the wreath product.
Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.
Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:
Then that container is refilled from the next oldest container, and that one in succession from the second-oldest, down to the youngest container, which is refilled with new product.
Then there is his resuscitation of the method of " water-culture ," and the application of it to the investigation of the problems of nutrition ; and further, his discovery that the starch-grains to be found in chloroplasts are the first visible product of their assimilatory activity.
Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set.
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