Why do we care about dual spaces?

Question

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra. I was wondering if anyone knew a nice intuitive motivation for the study of dual spaces and whether or not they "show up" as often as other concepts in linear algebra? Is their usefulness something that just becomes more apparent as you learn more math and see them arise in different settings? --- ### Edit I understand that dual spaces show up in functional analysis and multilinear algebra, but I still don't really understand the intuition/motivation behind their definition in the standard topics covered in a linear algebra course. (Hopefully, this clarifies my question)

Answer 1

Hi Giselle,

Dual spaces allow us to project/extend a vector space into a "mirror" vector space with all the linear functionals (addition and multiplication) from the original space. Dual spaces also allow us to determine the scalar product of a vector on itself, and evolve scalar descriptions for vector spaces. In other words:

Consider a vector space V over the space of a field F. The dual vector V* is defined as all the linear maps Φ: V → F.

Dual vectors make a "mirror dimension" of vectors V* which we call the (complex) conjugate of the vector. When we endow the dual space with addition and multiplication, the vector space V* becomes a vector space over F satisfying the same vector space properties as V.

Dual vectors allow us to find NORMS. Choose V = (a, b), where a, b ∈ C (complex numbers.) Then V* = (a*,b*), and VV* = V*V with |V| = √(V*V) = √(VV*)=√[(a,b)(a*,b*)]=√(aa*+bb*). If a and b are real, then we have the Pythagorean theorem, with |V| as the radius.

I Hope this helps.