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Let V!=(0) be a finite dimensional vector space over the complex numbers, and let A be a linear operator on V. Then V can be expressed as a direct sum of cyclic subspaces.

A sum of the elements from some set with constant coefficients placed in front of each. For example, a linear combination of the vectors x, y, and z is given by ax+by+cz, ...

The nth cubic number n^3 is a sum of n consecutive odd numbers, for example 1^3 = 1 (1) 2^3 = 3+5 (2) 3^3 = 7+9+11 (3) 4^3 = 13+15+17+19, (4) etc. This identity follows from ...

The sum of the reciprocals of roots of an equation equals the negative coefficient of the linear term in the Maclaurin series.

Diagonalize a form over the rationals to diag[p^a·A,p^b·B,...], where all the entries are integers and A, B, ... are relatively prime to p. Then Sylvester's signature is the ...

Given n mutually exclusive events A_1, ..., A_n whose probabilities sum to unity, then P(B)=P(B|A_1)P(A_1)+...+P(B|A_n)P(A_n), where B is an arbitrary event, and P(B|A_i) is ...

Every odd integer n is a prime or the sum of three primes. This problem is closely related to Vinogradov's theorem.

The radical line, also called the radical axis, is the locus of points of equal circle power with respect to two nonconcentric circles. By the chordal theorem, it is ...

An equation for a lattice sum b_3(1) (Borwein and Bailey 2003, p. 26) b_3(1) = sum^'_(i,j,k=-infty)^infty((-1)^(i+j+k))/(sqrt(i^2+j^2+k^2)) (1) = ...

As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (g(3)=9), that every "sufficiently large" ...