5.1, problem 7

The following are excerpts from a mail I sent to
a student in our class today, about problem 7 in
section 5.1:

Dear --------,

No, there is no mistake in the phrasing of the problem.
This is why I assigned the problem.  It is illuminating.  :-)
    _  _
Let u, v be in the vector space in problem 7.  By definition,
u, v are positive real numbers.  So their product uv  (as real
numbers) is again a positive real number.  So it is again in the
vector space.  
     _ _        __
So  (u,v) ----> uv   defines a function  "+"   from  V x V  to  V,
as described in class.  (Obviously this + has nothing to do with
addition of real numbers or addition of positive real numbers....)

Example:  3, 5 are in the vector space.  To emphasize that we
think of them as elements of the v. sp., write them in bold --

                 _   _
The vector sum   3 + 5   is the real number  15,   which of
course is positive, so is in the vector space.  So we write
it as a bold 15,

15 ,  to emphasize that we now think of it as a "vector" in a
vector space.

And if c is a real scalar and  v  is a pos. real number,
the scalar multiple
 _                                               c
cv  is  by definition the positive real number  v .

So e.g.  if c is the perfectly good real scalar  -1/2, and
v  is  7.3  ,  then
 _                     -1/2
cv  is by defn.   (7.3)     ,  i.e.,  1 over the nonnegative
square root of  7.3.  (Or rather, it is this positive real
number written in bold, since we think of it now as a "vector".)

You asked about what notation is ok to use, and what should
appear in a proof for problem 7 --

As for notation etc., just be careful with the definitions,
maybe use bold (as I suggested in class) where it is helpful,
and write down enough to convince the grader that you have
*checked* that the operations, as described, do indeed define 
functions "vector addition" and "scalar multpcn."  (I have
basically done this for you for free above, where I say
that certain things are positive reals.)

Then carefully verify that the 8 vector space axioms *do*
hold.  For this part, you are allowed to use familiar properties
of the "field" |R  of real numbers.  This should be easy.  The
hard part is "swallowing"  the initial definitions, as you have
noticed!   :-)

(In checking the axioms, don't pull the bonehead stunt so
many students do, of giving "examples".  The axioms are
supposed to hold for all vectors and scalars appearing
in them.  "Giving examples" has no place in proofs.
One can give a gazillion examples, and the proper reply is
"Fine.  Now what about the infinitely many cases you did 
not consider yet?")

You also asked for a sample solution to an earlier problem
in the 5.1 exercises.  I do not have my book at home....
As I recall, though, doing one of problems 1-6 does not
really help with problem 7.  If I remember, I will check
Tuesday and post something here if I think it helpful. This
mail already sort of wrecks problem 7, though.

The Chief