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In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
The splitting field of xq − x over Fp is the unique finite field Fq for q = pn. [2] Sometimes this field is denoted by GF ( q ). The splitting field of x2 + 1 over F7 is F49; the polynomial has no roots in F7, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4. [3]
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop: Input: matrices A and B.
The ordinals do not have unique factorization into primes under the natural product. While the full polynomial ring does have unique factorization, the subset of polynomials with non-negative coefficients does not: for example, if x is any delta number, then x 5 + x 4 + x 3 + x 2 + x + 1 = (x + 1) (x 4 + x 2 + 1) = (x 2 + x + 1) (x 3 + 1)
The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives.
Formal definition. The exterior algebra of a vector space over a field is defined as the quotient algebra of the tensor algebra by the two-sided ideal generated by all elements of the form such that . [6] Symbolically, The exterior product of two elements of is defined by.
Expanding a 1/ r potential. The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.
Infinite product. In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product. is defined to be the limit of the partial products a1a2 ... an as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge.