Opinions on Linear form

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In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars. In R, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the matrix product with the row vector on the left and the column vector on the right.  In general, if V is a vector space over a field k, then a linear functional f is a function from V to k that is linear:

f(\vec{v}+\vec{w}) = f(\vec{v})+f(\vec{w}) for all \vec{v}, \vec{w}\in V
f(a\vec{v}) = af(\vec{v}) for all \vec{v}\in V, a\in k.

The set of all linear functionals from V to k, Homk(V,k), forms a vector space over k with the addition of the operations of addition and scalar multiplication (defined pointwise).  This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space.  It is often written V or V′ when the field k is understood.


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