Physics 374
Fall 2005

HW Set 7
Problem 2

Well, this example can be shown to satisfy all the requirements for a linear space that we talked about in class.  A vector in such a space would be four dimensional.  For example, suppose I had 50 dollars, 100 Rials, 20 Pesos, and 200 acres.  My wealth vector would be (Subscript[W, 1 ])= (50,100,20,200).  Suppose two of my friends had the wealth vectors (Subscript[W, 2 ])= (10,500,0,100) and (Subscript[W, 3 ])= (20,400,50,200).

->Vector addition is associative...think of it like combining our assets:
	((Subscript[W, 1])+ (Subscript[W, 2])) + (Subscript[W, 3]) = (Subscript[W, 1])+ ((Subscript[W, 2]) + (Subscript[W, 3])) = (80,1000,70,500)
->There exists an identity vector "I"...think of it like someone with no wealth:
	I = (0,0,0,0)  so that (Subscript[W, 1]) + I = I + (Subscript[W, 1]) = (Subscript[W, 1])
->There exists an inverse for vector addition...think of it like combining your net wealth with someone who's in debt by the same amount you own.  For example, the additive inverse for (Subscript[W, 1]) would be -(Subscript[W, 1]) = (-50,-100,-20,-200).  Then we'd have (Subscript[W, 1]) + (-(Subscript[W, 1])) = I = (0,0,0,0).

Note how this is very much like adding vectors in 2-dimensional Cartesian space...the x and the y components don't "talk" to each other.  The non-interconvertibility of the currencies in the present example produces a  similar effect.

->Scalar multiplication is defined...think of putting three people with identical wealth together..you still get a vector in our space:
	3(Subscript[W, 1]) = (150,300,60,600).
->There exists a scalar multiplicative identity:
	1(Subscript[W, 1]) = (Subscript[W, 1])
->Scalar multiplication is associative...if I take 5 groups of my tri-combined people I get the same total wealth as taking 3 groups of 5-combined people:
	5(3(Subscript[W, 1])) = 3(5(Subscript[W, 1])) = 15(Subscript[W, 1])

Also, you can quickly convince yourself that
->Scalar multiplication and vector addition are distributive:
	(3 + 7) (Subscript[W, 1]) = 3(Subscript[W, 1]) + 7(Subscript[W, 1])

So, strictly speaking, this is an example of what a mathematician would call a "linear space".  It is spanned by four orthogonal unit-vector-like-things corresponding to dollars (1,0,0,0), rials (0,1,0,0), pesos (0,0,1,0) and acres (0,0,0,1).  Any wealth vector is a linear combination of these:
		W = a(1,0,0,0) + b(0,1,0,0) + c(0,0,1,0) + d(0,0,0,1)
Calling it a "vector space" brings in some nit-picky semantics. For most concerned-with-the-real-world physicists, these things like (Subscript[W, 1]) that I've been calling "vectors" don't have several properties that physicists usually ascribe to vectors.  For example, you can't really define a length for a "vector" like (Subscript[W, 1]).  How could you physically interpret something like \(TraditionalForm\`\@\(\((20  Dollars)\)\^2 + \((50\ Rials)\)\^2 + \((100  Pesos)\)\^2 + \((15  Acres)\)\^2\)\)?  You can't even add those elements together because they have different units!  Physicists also usually ascribe some notion of rotation to vectors.  I can rotate a  Cartesian vector like (Overscript[r,  ->])  =  
        6  (Overscript[x,  ∧])    -  7  (Overscript[y,  ∧])  by, say, 70 degrees counterclockwise and I'll get a new vector with new components that represent a specific "mixing" of the previous x and y components.  With the wealth "vectors", you can't really "rotate" them...what does it mean to "mix" dollars and rials?  They're non-interconvertible.

Solution by T . Bing