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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Discussion 4 -- Inner Products}
\centerline{April 21, 2004}
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\noindent
{\it Background}
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We have already used the dot product of vectors in ${\bf R}^n$
on a number of occasions in this course.  Recall that if 
${\bf x} = (x_1,x_2,\ldots,x_n)$ and ${\bf y} = (y_1,y_2,\ldots,y_n)$,
then ${\bf x}\cdot {\bf y}$ is the scalar:
$${\bf x}\cdot {\bf y} = x_1y_1 + x_2 y_2 + \cdots + x_n y_n.$$
Today we want to look at some properties of this operation
and its generalizations.
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\noindent
{\it Discussion Questions}
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\noindent
A)  At the start of multivariable calculus, you saw essentially that
all of the {\it geometry} of ${\bf R}^n$ ``comes from''
this dot product operation.   How do you use the dot product to measure the 
{\it distance} between two points $P_1$ and $P_2$ in ${\bf R}^n$?
How do you use the dot product to measure the 
{\it angle} between two vectors ${\bf x}$ and ${\bf y}$ in ${\bf R}^n$?
How do you tell when two vectors are {\it orthogonal} (perpendicular)?
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\noindent
B)  We want to take a slightly different ``angle'' in discussing
the dot product now.  Namely, we want to think of the dot product
as a {\it mapping} from one space to another.
What is the domain of the dot product mapping?
What is the target space of the dot product mapping?
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\noindent
C)  We also want to introduce an alternative notation for the 
dot product that is used in Chapter 4 of the text.  We will 
write 
$${\bf x}\cdot {\bf y} = \langle {\bf x}, {\bf y}\rangle = 
x_1y_1 + x_2 y_2 + \cdots + x_n y_n$$
Show using this notation that the dot product
has the following properties:
\item{1)}  (``bilinearity'')  For all ${\bf x}, {\bf y}, {\bf z}\in {\bf R}^n$,
and all $c \in {\bf R}$,
$\langle c{\bf x}+{\bf y}, {\bf z} \rangle = 
c\langle {\bf x}, {\bf z} \rangle + \langle {\bf y}, {\bf z} \rangle$
and 
$\langle {\bf x}, c{\bf y}+ {\bf z} \rangle = 
c\langle {\bf x}, {\bf y} \rangle + \langle {\bf x}, {\bf z} \rangle$.
Also, why is this property called ``bilinearity''?  How does it 
relate to the property of linearity for a mapping from one 
vector space to another?
\item{2)} (``symmetry'')  For all ${\bf x}, {\bf y}\in {\bf R}^n$,
$\langle {\bf x}, {\bf y}\rangle = 
\langle {\bf y}, {\bf x} \rangle $.
\item{3)} (``positive-definiteness'')  For all ${\bf x}\in {\bf R}^n$,
$\langle {\bf x}, {\bf x}\rangle \ge 0$,
and equals $0$ if and only if ${\bf x}$ is the zero vector.
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\noindent
D)  A set of vectors $S = \{{\bf x}_1, \ldots, {\bf x}_n\}$ is
said to be {\it orthogonal} if $\langle {\bf x}_i,{\bf x}_j\rangle = 0$
for all pairs $i\ne j$.  $S$ is said to be {\it orthonormal} if 
$S$ is orthogonal, and in addition 
$\|x_{i}\| = \sqrt{\langle {\bf x}_i, {\bf x}_i\rangle} = 1$
for all $i$.
\item{1)}  Show that an orthogonal set of nonzero vectors is
linearly independent.  
\item{2)}  Show that if $\beta = \{{\bf x}_1,\ldots, {\bf x}_n\}$
is any orthonormal basis for ${\bf R}^n$, then the scalars in
the linear combination expressing an arbitrary vector
${\bf x}$ in terms of $\beta$:
$${\bf x} = a_1 {\bf x}_1 + \cdots + a_n {\bf x}_n$$
are given by $a_i = \langle {\bf x}, {\bf x}_i\rangle$
for all $i$.
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\noindent
E)  The three properties in question C form the definition of a more general
concept called an {\it inner product} on a vector space $V$.
Any mapping $\langle\ ,\ \rangle : V \times V \to {\bf R}$
satisfying them is called an inner product on $V$.
\item{1)}  Show that the mapping 
$\langle\ ,\ \rangle : {\bf R}^2 \times{\bf R}^2 \to {\bf R}^2$
defined by 
$$\langle (x_1,x_2),(y_1,y_2) \rangle = \pmatrix{x_1 &x_2}
\pmatrix{2&1\cr 1&1\cr} \pmatrix{y_1\cr y_2}$$
(matrix product) is an inner product on ${\bf R}^2$ (different from the dot 
product!)  
\item{2)} (An inner product from calculus -- if you're taking
the ODE/Applied Math sequence next year, you'll see this example
again!)  Let $V$ be the vector space of all continuous functions
on the unit interval $[0,1]$.  Show that
$$\langle f,g\rangle = \int_0^1 f(x)g(x)\ dx$$
defines an inner product on $V$.  (Note: the hardest part here
is to show that if $f$ is not the zero function, then 
$\int_0^1 (f(x))^2\ dx > 0$.  You'll need to think about the definition
of the definite integral and the properties of continuous
functions here.)
\item{3)}  Let $V = M_{2\times 2}({\bf R})$  be the vector
space of $2 \times 2$ matrices.  Show that 
$$\langle A,B\rangle = Tr(A^tB)$$
(that's the trace of the matrix product of $A$ transpose 
and $B$) defines an inner product.  
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{\it Assignment}
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Group write-ups due in class on Monday, April 25.
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