Dual Vector Space

In mathematics any vector space v has a corresponding dual vector space or just dual space for short consisting of all linear functionals on v together with the vector space structure of pointwise addition and scalar multiplication by constants.

Dual vector space. In the abstract vector space case where dual space is the algebraic dual the vector space of all linear functionals a vector space is isomorphic to its algebraic dual if and only if it is finite dimensional. It is a vector space because such columns can be multiplied by rational scalars and added to get more of the same. So in a sense i am associating a number with a function.

Bill dubuque gives a nice argument in a sci math post see google groups or mathforum. There are more rows than there are columns. N be a basis of v.

The dual vector space to a real vector space is the vector space of linear functions denoted. For each i 1 n de ne a linear functional f. This dual space is not like the original vector space at all.

In the dual of a complex vector space the linear functions take complex values. The dual space of v denoted by v is the space of all linear functionals on v. V is dual of v and hence collection of all maps from v to f and this collection forms a vector space which i think is easily proved.

The space of linear maps from v to f is called the dual vector space denoted v. Dual vector spaces defined on finite dimensional vector spaces can be used for defining tensors. Suppose that v is nite dimensional and let v.

In either case the dual vector space has the same dimension as. J ˆ 1 if i j 0 if i 6 j and then extending f. V f by setting f.