Dual Vector Space

The dual space consists of rows with infinitely many rational components chosen arbitrarily.

Dual vector space. It is a vector space because such columns can be multiplied by rational scalars and added to get more of the same. 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. Suppose that v is nite dimensional and let v.

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. Each vector in this double dual vector space is a linear transformation from v to f. The map is an isomorphism if and only if the space is finite dimensional. The dual space of v denoted by v is the space of all linear functionals on v.

In either case the dual vector space has the same dimension as. Dual vector spaces defined on finite dimensional vector spaces can be used for defining tensors. Dual spaces and linear functionals in this video i introduce the concept of a dual space which is the analog of a shadow world version but for vector sp. For each i 1 n de ne a linear functional f.

In mathematics any vector space v has a corresponding dual vector space or just dual spacefor short consisting of all linear functionals on v. There are more rows than there are columns. N be a basis of v. The exterior product of two vectors u and v denoted by u v is called a bivector and lives in a space called the exterior square a vector space that is distinct from the original space of vectors.

Bill dubuque gives a nice argument in a sci math post see google groups or mathforum. 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. 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. J ˆ 1 if i j 0 if i 6 j and then extending f.

So in a sense i am associating a number with a function. The space of linear maps from v to f is called the dual vector space denoted v.