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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 ...

A subset M of a Hilbert space H is a linear manifold if it is closed under addition of vectors and scalar multiplication.

Scalar multiplication refers to the multiplication of a vector by a constant s, producing a vector in the same (for s>0) or opposite (for s<0) direction but of different ...

The divergence of a vector field F, denoted div(F) or del ·F (the notation used in this work), is defined by a limit of the surface integral del ·F=lim_(V->0)(∮_SF·da)/V (1) ...

The curl of a vector field, denoted curl(F) or del xF (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at ...

The orthogonal decomposition of a vector y in R^n is the sum of a vector in a subspace W of R^n and a vector in the orthogonal complement W^_|_ to W. The orthogonal ...

A vector norm defined for a vector x=[x_1; x_2; |; x_n], with complex entries by |x|_1=sum_(r=1)^n|x_r|. The L^1-norm |x|_1 of a vector x is implemented in the Wolfram ...

The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Norms exist for complex numbers ...

Given a contravariant basis {e^->_1,...,e^->_n}, its dual covariant basis is given by e^->^alpha·e^->_beta=g(e^->^alpha,e^->_beta)=delta_beta^alpha, where g is the metric and ...

The l^2-norm (also written "l^2-norm") |x| is a vector norm defined for a complex vector x=[x_1; x_2; |; x_n] (1) by |x|=sqrt(sum_(k=1)^n|x_k|^2), (2) where |x_k| on the ...