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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Discussion 1 -- Extending Vector Operations to ${\bf R}^n$}
\centerline{September 7, 2007}
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{\it Background}
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The definitions of ${\bf R}^2$ and ${\bf R}^3$ can be extended to give
$n$-dimensional spaces ${\bf R}^n$ for {\it any} $n \ge 1$:
$${\bf R}^n = \{(a_1,\ldots,a_n) : a_i \in {\bf R} {\rm\ for\ all\ } i = 1,\ldots,n\}.$$
We will see today that many of the vector operations we have considered
in 2 and 3 dimensions can be extended to these spaces as well.  But first we want to consider
a question that might be at the forefront of your minds!  {\it Why take $n > 3$?}
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{\it Discussion Questions}
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A)  ${\bf R}^2$ has the obvious interpretation of the coordinate plane
from geometry.  ${\bf R}^3$ can be seen as a mathematical model of the 
(apparently) 3-dimensional physical world we live in.  Although the 
mathematical definition of ${\bf R}^4$ or ${\bf R}^n$ more generally
presents little difficulty, it is still natural to ask {\it why} might
the mathematical object ${\bf R}^n$ for $n \ge 4$ be useful?   Here
are some examples:
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\item{1)}  In physics, when considering objects in motion, we often
need to know not just {\it where} an object is, but {\it when} it is
there.  If we think of the ``complete coordinates'' of an object
as its three spatial coordinates, and the current time, what 
space ${\bf R}^n$ contains all the possible ``complete coordinate
vectors?''
\item{2)}  Again in physics, suppose we are interested in the 
configurations of {\it several objects} moving in space simultaneously. 
For instance, say we have the configuration of the center of the Sun,
the center of the Earth, and the center of the Moon.  To describe 
each center's position, we need three spatial coordinates, so how
many coordinates do we need to specify the whole configuration
(Sun, Earth, Moon)?   What 
space ${\bf R}^n$ contains all the possible ``configuration vectors?''
\item{3)}  In economics, suppose we wanted to represent the current
state of the world economy by giving the GNP's of each of the 
190 or so countries in the world.  What 
space ${\bf R}^n$ contains all the possible ``world economy state vectors?''
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\item{} All of these are realistic examples of the way spaces
${\bf R}^n$ for $n \ne 2,3$ are used to study situations mathematically(!)
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B)  A {\it vector sum} operation can be defined in ${\bf R}^n$ for any $n$:
$$(a_1,\ldots,a_n) + (b_1,\ldots,b_n) = (a_1 + b_1, \ldots, a_n + b_n).$$
Each vector in ${\bf R}^n$ can be written in one and only one way
as a linear combination of a collection of vectors analogous to 
the {\it standard basis} ${\bf i,j,k}$ in ${\bf R}^3$.  What are 
these vectors, and how does this work?
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C)  A {\it dot product} can be defined for vectors in ${\bf R}^n$
by 
$${\bf a} \cdot {\bf b} = (a_1,\ldots,a_n) \cdot (b_1,\ldots,b_n) = a_1 b_1 +  \cdots + a_n b_n.$$
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\item{1)}  Show that ${\bf a} \cdot {\bf a} \ge 0$ for all ${\bf a}$ in
${\bf R}^n$, and ${\bf a} \cdot {\bf a} = 0$ only if ${\bf a} = (0,\ldots,0)$.
\item{2)}  Show that ${\bf a} \cdot {\bf b} = {\bf b} \cdot {\bf a}$ for
all ${\bf a}, {\bf b}$ in ${\bf R}^n$.
\item{3)}  Show that
$${\bf a} \cdot ({\bf b} + {\bf c}) = {\bf a}\cdot {\bf b} + {\bf a}\cdot {\bf c}$$
for all ${\bf a}, {\bf b}, {\bf c}$ in ${\bf R}^n$.
\item{4)}  Say how to define a norm $\|{\bf a}\|$ for vectors ${\bf a} \in {\bf R}^n$.
\item{5)}  Show that the Cauchy-Schwarz inequality:
$$|{\bf a}\cdot {\bf b}| \le \|{\bf a}\| \|{\bf b}\|$$
holds for all ${\bf a},{\bf b}$ in ${\bf R}^n$. (Hint:  it suffices to show
$$({\bf a}\cdot {\bf b})^2 \le \|{\bf a}\|^2 \|{\bf b}\|^2;$$
do you see why?)
\bye
