Concepts + an application:

A general strategy in mathematics is the following:  We take some useful object (e.g. numbers, or matrices,
or vectors), then list its properties.  Then, we search for different objects that have the same properties.
Take for instance R^n.  Its algebraic properties are listed on page 27.

Pretty much everything we will learn about R^n in this course will be derived from
the properties listed in the box on page 27.  Why does that matter?  The reason is
as follows:  If some other object we haven't even thought about also satisfies the
algebraic properties on page 27,  then   everything we've learned about R^n will
work for that object as well!
Because of this, theorems in linear algebra will have many applications that the inventors
of those theorems never imagined.  The people that produced these theorems could never
have imagined that these results would be used by the error-correction-techniques that
are vital for the proper functioning of modern phones.

Here is the basic idea:  The cellphone tower sends a string of 0's and 1's to your phone:
   ....0010011100....   (millions of 0's and 1's)
but there will be some errors, and they will need to be corrected.  The simplest error correction
is as follows:    Replace each "0" with "000" and replace each "1" by "111".
If one error occurs, say "000" --> "001" then the receiver can detect+correct this error.

There are two big problems with the "simplest error correction":
	1) The data gets 3 times longer
	2) As little as 2 errors is all that it takes for things to go wrong
	   "0" encodes to "000", with 2 errors this becomes "011", which decodes to "1"

We can do better. This string of millions of 0's and 1's,  lets split them into vectors with 1000 entries
each:  ....0010011100.... --->  v1 v2 v3 ...
where each vector v1, v2, v3,... is in  {0,1}^1000
(an element of {0,1}^1000 is a vector with 1000 entries, and each entry is 0 or 1).

Now the encoding will go from {0,1}^1000 to {0,1}^1100.
So the encoding turns a vector with 1000 entries into a vector with 1100 entries.
Those 1100 entries are then transmitted from the cellphone tower to your phone (some of those
entries can change along the way due to interference).
Your phone receives those 1100 entries (some of which are wrong), it corrects the errors
and retrieves the original 1000 entries error-free.

Advantages:
	1) The encoding only makes the data 10% longer
	2) We can correct far more errors. In the simplest method, 2 errors is too much.
           But because we added 100 entries to each vector, we can correct about 40 errors!

So working with vectors with 1000 entries and 1100 entries, we only need to add 10% data and
we can correct a large number of errors.  This makes the data-transfer fast and extremely
reliable.   In fact, due to the far greater reliability, we can drastically increase the
data-transfer-rate and still get reliable results. This greatly speeds up the overall
communication. This is what allows you to watch youtube in HD directly on your phone in
real time.

Remains to do:  to explain how exactly a cell phone tower can encode a vector in {0,1}^1000
with a vector in {0,1}^1100, then transmit that to your phone, in such a way that it can
correct numerous errors that could have occurred during the transmission.

For this, we'll need the concepts from this course, vector space, subspace, basis, etc.