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A projective space is a space that is invariant under the group G of all general linear homogeneous transformation in the space concerned, but not under all the ...

D_(KY)=j+(sigma_1+...+sigma_j)/(|sigma_(j+1)|), (1) where sigma_1<=sigma_n are Lyapunov characteristic exponents and j is the largest integer for which ...

Let V and W be vector spaces over a field F, and let T:V->W be a linear transformation. Assuming the dimension of V is finite, then dim(V)=dim(Ker(T))+dim(Im(T)), where ...

The vector space generated by the columns of a matrix viewed as vectors. The column space of an n×m matrix A with real entries is a subspace generated by m elements of R^n, ...

A complex manifold is a manifold M whose coordinate charts are open subsets of C^n and the transition functions between charts are holomorphic functions. Naturally, a complex ...

The vector space generated by the rows of a matrix viewed as vectors. The row space of a n×m matrix A with real entries is a subspace generated by n elements of R^m, hence ...

An object is said to be self-similar if it looks "roughly" the same on any scale. Fractals are a particularly interesting class of self-similar objects. Self-similar objects ...

The degree (or relative degree, or index) of an extension field K/F, denoted [K:F], is the dimension of K as a vector space over F, i.e., [K:F]=dim_FK. If [K:F] is finite, ...

A representation phi of a group G is faithful if it is one-to-one, i.e., if phi(g)=phi(h) implies g=h for g,h in G. Equivalently, phi is faithful if phi(g)=I_n implies g=e, ...

Regardless of where one white and one black square are deleted from an ordinary 8×8 chessboard, the reduced board can always be covered exactly with 31 dominoes (of dimension ...