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Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
The Casimir element for B is the element of the universal enveloping algebra given by the formula Ω = ∑ i = 1 n X i X i . {\displaystyle \Omega =\sum _{i=1}^{n}X_{i}X^{i}.} Although the definition relies on a choice of basis for the Lie algebra, it is easy to show that Ω is independent of this choice.
The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
For a field F, the generalized special unitary group over F, SU(p, q; F), is the group of all linear transformations of determinant 1 of a vector space of rank n = p + q over F which leave invariant a nondegenerate, Hermitian form of signature (p, q). This group is often referred to as the special unitary group of signature p q over F.
The nullity of M is given by m − n + c, where, c is the number of components of the graph and n − c is the rank of the oriented incidence matrix. This name is rarely used; the number is more commonly known as the cycle rank, cyclomatic number, or circuit rank of the graph. It is equal to the rank of the cographic matroid of the graph.
The names "caffeine" and "3,7-dihydro-1,3,7-trimethyl-1H-purine-2,6-dione" both signify the same chemical compound.The systematic name encodes the structure and composition of the caffeine molecule in some detail, and provides an unambiguous reference to this compound, whereas the name "caffeine" simply names it.
Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Temperature gradient: No standard symbol K⋅m −1: ΘL −1: Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer
The (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous space SO + (1, 3) / SE(2) is the momentum space of a massless particle; geometrically, this Kleinian geometry represents the degenerate geometry of the light cone in Minkowski spacetime.