Why do we learn Dual Space so early?

[Vorde]

I'm in a Second Course in Linear Algebra this semester, and we've just been introduced to the idea of a dual space, dual vectors and briefly to a double dual space. I completely understand how all of these things work and how they're defined, but I don't understand why we care.

I've been doing some research on them online, and I understand some of the broader ideas which they are used for, but everything which I can find relating to dual spaces seems to be at a higher level than what we are dealing with in this class.

So my question is then, why do we learn about dual spaces as early as we do, and why do we care about them now?

Thank you

 

[Number Nine]

Several (possible) reasons. The first is that it's a very straightforward concept, and there's nothing to be lost by introducing it early, especially given how common it is. Second: if this is a second course in linear algebra, you're probably going to be talking about inner product spaces, where the inner product of a vector with a fixed vector v can be viewed as a linear functional, which gives a natural connection to the dual space.

 

[lurflurf]

The second course? Why were dual spaces introduced so late? I wish I could find again a comparison I once read of the page numbers on which dual spaces were introduced in the popular linear algebra books. The best books introduced dual space earliest. There are several advantages to doing so. Dual spaces are a natural and clear way to think about linear algebra. When dual spaces are introduced later students often have to unlearn some of what they learned earlier. Dual spaces become even more important when working over structures other than complex numbers or when doing multilinear algebra.

 

[Vorde]

Thank you guys, I think I understand it now.

I have a different, but related question, and I'll ask it here so as to avoid clogging up the forum with another dual space question.

After wrestling with it for a while, I think I have a comfortable, intuitive sense of how and why a dual mapping works (a transpose of a mapping). However, in order to make myself more comfortable I've tried to give myself a concrete example:

Take the matrix A =
(3, 1, 4)
(2, 2, 1)
Take the vector x = [1, 1, 1] $\in$ R3
And take the linear functional $\phi$ $\in$ R2 s.t. $\phi$([a,b]) = a+b

Ax = [8, 5], and so $\phi$(x) = 13

However, I'm stuck on trying to move the opposite way with the transpose mapping.
My problem seems to be that I am unable to figure out how to represent $\phi$ using coordinates of the dual basis of R2. I understand theoretically how to produce a dual basis, but I am unable to actually write down coordinates for it.

Am I on the right track? Once I can represent phi as coordinates I would multiply that coordinate vector by the matrix transpose and then convert back to a functional from R3 to R. Is this correct? And how do I do so? Thank you.


Like all things annoying:

The moment I posted this I figured it out, thanks anyways though.

 

[mathwonk]

we study spaces in two ways: 1) maps of standard spaces into them, 2) maps of them into standard spaces.

if V is a vector space over R, then the space

1) of linear maps R-->V is isomorphic to V.

2) of linear maps V-->R is the dual space.

i.e. it is often valuable to understand a space by studying maps on that space.

sorry i know this isn't totally persuasive.

but think about it: what else are you going to do?