\begin{problem}{MATH 294}{\spr{83}}{FINAL}{3}{}
\prob{Consider the vector space of functions over the interval $0
\leq x \leq 1 $ and the inner product \[ (f,g) = \int^1_0 f(x)
g(x) d x.
\] Find an orthogonal basis for the subspace spanned by $1, e^x, 2 e^x , e^{-x}.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{87}}{\pr{3}}{6}{}
\prob{Find an element of the vector space $V$ which is functions
of the form $a e^{-t} + b e^{-2 t}$ (where $a$ and $b$ are
arbitrary constants) which is orthogonal to $E^{-t}$. Use the
following inner product: $<f(t),g(t)> \equiv \int_0^\infty f(t)
g(t) d t.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{5 MAKEUP}{}
\prob{Suppose $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n$ is an
orthonormal basis for $\Re^n$, and $\vec{v}$ is a vector in
$\Re^n$ such that $<\vecv, \vecv_i> = 0$ for all $i =
1,2,\ldots,n.$ (Here $<v,w>$ denotes the standard inner product
in $\Re^n$.) Then what can you conclude about $\vecv$? Why?
(Hint: write $\vecv$ as a linear combination of the basis
elements $\vecv_1, \ldots, \vecv_n,$ then apply the condition
$<\vecv, \vecv_i> = 0.$)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{6 MAKE-UP}{}
\prob{With the inner product $<f,g> = \frac{1}{\pi}
\int_{-\pi}^{\pi} f(t) g(t) d t$ in the vector space of
continuous functions defined on $[-\pi, \pi]$ what is $||a \sin x
+ b \cos x ||?$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{FINAL}{8}{}
\prob{Let $<f,g> = \int_0^1 f(t) g(t) d t$ be an inner product on
$C_\infty[0,1]$. Determine: } \probpart{the component of $f(x) =
1+x $ along $g(x) = x^2.$} \probpart{the component of $f(x)$
perpendicular to $g(x)$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SUMMER 1989}{\pr{2}}{3}{}
\prob{Consider the vector space $C[0,1]$ of continuous functions
over the interval $0 \leq t \leq 1$ and inner product \[ <f,g> =
\int_0^1 f(t) g(t) d t. \]} \probpart{Show that $\set{1,t,t^2}$
is a set of linearly independent vectors in $C[0,1].$}
\probpart{Find an orthogonal basis for the subspace of $C[,1]$
spanned by $1,t,4 t, t^2.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{89}}{\pr{3}}{4}{}
\prob{Let $C[-1,1]$ denote the continuous real-valued functions
on $[-1,1]$, and let $W$ be the following subspace thereof: \[ W
= \set{c_1 + c_2 t + c_3 t^4 : c_1, c_2, c_3 \mbox{ real numbers
}}.
\]} \probpart{Show that $W$ is three dimensional.} \probpart{For functions $p(t), q(t)$ i $W$, introduce the following inner product: \[ <p,q> = \int^1_{-1} p(t) q(t) d t. \] Find an orthogonal basis for $W$ which contains the function $p(t) \equiv 1$ as one element.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{FINAL}{5}{}
\prob{In $V_4$, $\vec{u} = \vtfour{1}{0}{1}{1}$ and $\vec{v} =
\vtfour{-4}{3}{2}{1}$} \probpart{Using the standard inner
product\[ (\vec{u}, \vec{v}) = \sum^4_{i=1} u_i v_i \] find the
length of $\vec{u}$ and determine whether the angle between
$\vec{u}$and $\vec{v}$ is greater or less than 90 degrees.}
\probpart{Using the nonstandard inner product \[ (\vec{x},\vec{y})
= \sum^4_{k=1} k x_k y_k = x_1 y_1 + 2 x_2 y_2 + 3 x_3 y_3 + 4 x_4
y_4
\] find the length of $\vec{u}$and determine whether the angle between $\vec{u}$ and $\vec{v}$ is more or less than 90 degrees.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{FINAL}{5}{}
\prob{Given the set of functions \[ \set{1,x,x^2,x^3} \mbox{ with
} -1 \leq x \leq 1
\] and the inner product $\inp{f}{g}{-1}{1}{x}$} \probpart{Find an orthogonal basis $\brc{w_1(x), w_2(x), w_3(x), w_4(x)}$ of the space spanned by the functions $1, x, x^2 \and x^3.$ Use the Schmidt orthogonalization
procedure.}\probpart{Given $\phi(x) = 1+ 2x + 3x^2 \and \phi(x) =
c_1 w_1(x) + c_2 w_2(x) + c_3 w_3(x) + c_4 w_4(x), $ find $c_2$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{97}}{FINAL}{8}{}
\prob{You are given a vector space $V$ with an inner product
$<,>$ and an orthogonal basis
\renewcommand*{\bv}[1]{\vec{b}_{#1}}$B = \set{\bv{1}, \bv{2},
\bv{3}, \bv{4}, \bv{5}}$ for which $||\vec{b}_|| = 2, i =
1,\ldots , 5$. Suppose that $\vec{v}$ is in $V$ and \[
<\vec{v},\vec{b}_1> = <\vec{v},\vec{b}_2> = 0
\] \[ \and <\vec{v},\vec{b}_3> = 3, <\vec{v},\vec{b}_4> = 4, <\vec{v},\vec{b}_5> = 5\] Find the coordinates of $\vec{v}$ with respect to the basis $B$ i.e. find $c_1, c_2, c_3, c_4, c_5$ such that \[ \vec{v} = c_1 \vec{b}_1 + c_2 \vec{b}_2 +c_3 \vec{b}_3 +c_4 \vec{b}_4 +c_5 \vec{b}_5\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{FINAL}{7}{}
\prob{Consider the subspace \[ W = span\set{1,t}, \for 0 \leq t
\leq 1,
\] equipped with the inner product \[ \inp{f}{g}{0}{1}{t}. \] Find the best approximation to the function $f(t) = e^t$ in $W$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{FINAL}{7}{}
\prob{Regard $P^2$ as a subspace of $C[-1,1]$ and construct an
orthogonal basis from the standard basis $E = \set{1, t, t^2}$
using the inner product \[ \inp{p}{q}{-1}{1}{t}. \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{FINAL}{8}{}
\prob{Consider $P^2$ to be a subspace of $C[-1,1]$}
\probpart{Check that $\set{1,t,t^2 - \frac{2}{3}}$ is an
orthogonal basis for this subspace with respect to the inner
product \[ <f,g> \equiv f(-1)g(-1) + f(0)g(0) + f(1)g(1).\]}
\probpart{Determine the beat second order polynomial
approximation, $a_0, a_1 t + a_2 t,$ to the function $e^t$ with
respect to this inner product.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{\pr{3}}{2}{}
\prob{For all the questions below use the vector space
$C[-\pi,\pi]$, the set of all continuous functions between $-\pi
\and \pi$, and the inner product: \[ \inp{f}{g}{-\pi}{\pi}{t} \]
[Hint $\int^{\pi}_{-\pi} \sin^2 (t) d t = \int^{\pi}_{-\pi}
\cos^2 d t = \pi$ .]} \probpart{What is the "distance" between
the function $t$ and the function 1?} \probpart{Do the three
functions $\set{1, \sin(t), \cos(t)}$ form an orthogonal basis
for a subspace of $C[-\pi, \pi]$?} \probpart{What value should
$A$ have if $A \sin(t)$ is to be the best possible fit to the
function $t$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{\pr{2}}{5}{}
\prob{Consider $C^0([1,3])$. Is the constant function $g(x) = 1
(1 \leq x \leq 3)$ a unit vector? Find the orthogonal projection
$w(x)$ of $f(x) = \frac{1}{x} (1 \leq x \leq 3)$ onto the span of
$g(x)$. \[ [\mbox{Here } f \cdot g = \int^3_1 f(x) h(x) d x] \] }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{5}{}
\prob{In the space of continuous functions of $x$ in the interval
$1 \leq x \leq 2$ one may define an inner product (other than the
usual one) as follows: $f \cdot g = \int_1^2 \frac{1}{x} f(x)
g(x) d x.$ } \probpart{Using this inner product of $f$ and $g$
find \[ ||f|| \mbox{ (norm of $f$ ) if } f(x) = \sqrt{x} \for 1
\leq x \leq 2,
\]} \probpart{Determine the real constants $a,b, c$ which make the set $\set{a, b+c x}$ orthonormal (leave $\ln 2$ as is in answers).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{}{}{3 PRACTICE}{}
\prob{Let $V = C^0 [-1,1]$ with the inner product \[ f \cdot g =
\int^1_{-1} f(x) g(x) d x.
\]} \probpart{Find an orthogonal basis for the space spanned by the functions $f_1(x) = 1, f_2 (x) = x, f_3 (x) =
x^2.$}\probpart{Find the orthogonal projection of $x^3$ onto the
subspace spanned by the functions $1,x,x^2.$}
\end{problem}
