[Other] Conceptually, what is a Linear Functional?

[alakazam]

I know this isn't the best fit for this thread in terms of categories, but I'm sure I'm not the only NT who has struggled to find a good explanation of this. I figure I'm more likely to find someone here with the same troubles I've had than anywhere else (and more likely to find someone here that's heard of a linear functional).


I can't find a good explanation of this anywhere.

Every site I see tells me, implementation-ally, what it is. For example, Wolfram MathWorld says: "A linear functional on a real vector space is a function T : V -> R, which satisfies the following properties..."

However, it just looks like a [linear] function...


I see that, in one case (I think...), a 'linear functional' could be defined as T(a,b) = 2a + b so I could say "a linear functional is a function which can be easily represented by a vector; then, applying this function to another vector is equivalent to an inner product defined as the dot product"

However, in a number of homework problems, they define something convoluted as seen in the solutions for 6.25 here: http://www2.engr.arizona.edu/~gehm/501/files/solutions/HW4 Solutions.pdf


So, what IS a linear functional at is core? There HAS to be a simple explanation as to what it is that causes the details/notation to fall into place. However, everybody seems to work the other way around: starting with the muddy details and hoping the idea pops out of their butts.

 

[Rasofy]

I think this question is best suited to intpforum !

 

[thisGuy]

From Wikipedia: In advanced mathematics, a linear function means a function that is a map between two vector spaces that preserves vector addition and scalar multiplication.

Does this not make sense? Variations of this definition are used everywhere (from signals about a cellphone to quantum physics). Think of it as a way to convert from one vector space to a different vector space - thats essentially how it is used.

In one dimensional space you have the variable x. In two dimensional space, you have (x,0). They both mean the same thing to you - x. But they are represented (written) differently because they are in different spaces. Now how do you mathematically go from x to (x,0)? The answer to this will be a linear function.

Thats the very basics of it. But you won't be able to apply this definition anywhere. For pretty much every purpose you (they) need to define a 'from' space and a 'to' space; then you decide how to go from 'from' to 'to'. Then you decide if addition in 'from' space is same as addition in 'to' space and whether multiplying things in the 'from' space by a constant is same as multiplying things in 'to' space by a constant (can be a different constant - but you already knew that!).

For the addition condition, you may need two points in one space that you map to the other space.

Thats how I see it anyway. Its how you 'map' things in mathematics. But maths requires that everything be rigours; so they include they two conditions - if these match then your map is valid (some crazy genius guys must've come up with these a long time ago. just be happy you don't use THAT method and that you have these two simple conditions to figure it out.)

I do understand the difficulty you are having. When my prof used to do it, I thought it was the easiest thing I'd ever seen. But then I couldn't do it on my own - it was too simple a concept.

 

[Halla74]

y = mx + b

OR

Ax + By = C

To convert between them a nice example is here: http://mathcentral.uregina.ca/QQ/database/QQ.09.04/farzana1.html

-Alex