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Suppose that V={(x_1,x_2,x_3)} and W={(x_1,0,0)}. Then the quotient space V/W (read as "V mod W") is isomorphic to {(x_2,x_3)}=R^2. In general, when W is a subspace of a ...

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 ...

There are a couple of versions of this theorem. Basically, it says that any bounded linear functional T on the space of compactly supported continuous functions on X is the ...

Given a map f:S->T between sets S and T, the map g:T->S is called a right inverse to f provided that f degreesg=id_T, that is, composing f with g from the right gives the ...

A formula which transforms a given coordinate system by rotating it through a counterclockwise angle Phi about an axis n^^. Referring to the above figure (Goldstein 1980), ...

Every smooth nonzero vector field on the 3-sphere has at least one closed orbit. The conjecture was proposed in 1950 and proved true for Hopf maps. The conjecture was ...

A seminorm is a function on a vector space V, denoted ||v||, such that the following conditions hold for all v and w in V, and any scalar c. 1. ||v||>=0, 2. ||cv||=|c|||v||, ...

A Lie algebra g is solvable when its Lie algebra commutator series, or derived series, g^k vanishes for some k. Any nilpotent Lie algebra is solvable. The basic example is ...

Let V be a real vector space (e.g., the real continuous functions C(I) on a closed interval I, two-dimensional Euclidean space R^2, the twice differentiable real functions ...

Two points z and z^S in C^* are symmetric with respect to a circle or straight line L if all circles and straight lines passing through z and z^S are orthogonal to L. Möbius ...