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Here different solutions are identified if they are isomorphic (that is, geometrically the same). Moduli spaces can be thought of as giving a universal space of parameters for the problem. For example, consider the problem of finding all circles in the Euclidean plane up to congruence.
The space of all functions from X to V is commonly denoted V X. If X is finite and V is finite-dimensional then V X has dimension |X|(dim V), otherwise the space is infinite-dimensional (uncountably so if X is infinite). Many of the vector spaces that arise in mathematics are subspaces of some function space. We give some further examples.
Prior to the 1940s, algebraic geometry worked exclusively over the complex numbers, and the most fundamental variety was projective space. The geometry of projective space is closely related to the theory of perspective, and its algebra is described by homogeneous polynomials. All other varieties were defined as subsets of projective space.
More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, S 6. This is the set of all points in seven-dimensional space (Euclidean) that are a fixed distance from the origin. This constraint reduces the number of coordinates ...
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
A rearrangement of the entries of a banded matrix which requires less space. Sparse matrix: A matrix with relatively few non-zero elements. Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms. Symmetric matrix: A square matrix which is equal to its transpose, A = A T (a i,j = a j,i ...