The dual space of a vector space , sometimes denoted , is the vector space of all linear forms over with the obvious addition and scalar multiplication operations defined.
Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the , and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.
Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case
One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space .
In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:
Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector as:
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In mathematics, particularly in functional analysis and linear algebra, the concept of the **dual space** is important in studying vector spaces and linear maps. ### Definition Given a vector space \( V \) over a field \( F \) (commonly the real numbers \( \mathbb{R} \) or complex numbers \( \mathbb{C} \)), the **dual space** \( V^* \) is defined as the set of all linear functionals on \( V \).