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\begin{center}
\bf Some review questions: inner product spaces, eigenvalues
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\vspace{0.2in}

\problemno{1}
Let $V$ be an inner product space.
\par\noindent{\bf(a):\ } State precisely what it means for the inner product
to be {\it linear in the first variable}.
\par\noindent{\bf(b):\ } State precisely what it means for the inner product
to be {\it linear in the second variable}.
\par\noindent{\bf (c):\ } Let $u,v,w \in V$.  We know that
$$\inn{u,v} = 2,\ \inn{v,w} = -3, \inn{u,w} = 5,$$
$$\norm{u} = 1,\ \norm{v} = 2,\ \norm{w} = 7.$$%
Compute $\inn{u+v,v+w}$.

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\problemno{2}
Let $A$ be an $n \times n$ matrix with eigenvalues 
$\vec \lambda1n$.  What statement about the eigenvalues is equivalent
to saying that $A$ is invertible?

\vspace{0.1in}

\problemno{3}
Find the coordinates of $\BRMAT{1\cr-1\cr2}$ \WRT\ the basis
$$\BRMAT{-3/5\cr4/5\cr0},\ \BRMAT{4/5\cr3/5\cr0},\ \BRMAT{0\cr0\cr1}$$%
for $\R^3$.  [{\sc Hint:}\ What is special about this basis?]

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\problemno{4}
Equip $\R^3$ with the inner product given by
$$\inn{a,b}\ =\ a_1b_1 + 2a_2b_2 + 4a_3b_3.$$%
Let $p = \BRMAT{1\cr2\cr3}$, $q = \BRMAT{3\cr2\cr1}$.  With respect to the
given inner product, what is the distance between $p$ and $q$?

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\problemno{5}
Let $V$ be an inner product space with basis $v_1, v_2, v_3, v_4$.  You are
in the middle of applying the Gram-Schmidt process to find an
orthonormal basis $w_1, w_2, w_3, w_4$ for $V$: you have already found
$w_1, w_2, w_3$.  
\vspace{0.1in}
\par\noindent{\bf(a):\ } If $x,u \in V$, and $\norm{u} = 1$, give a formula for 
the {\it component\/} of $x$ in the direction $u$.
\par\noindent{\bf(b):\ }
Explain {\it precisely\/} how you compute $w_4$.  

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\problemno{6} 
Let $A = \BRMAT{1&a\cr0&1}$, where $a$ is nonzero.  Show that
$A$ is not diagonalizable.

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\problemno{7} 
Let $V$ be the vector space of all continuous functions from
$[0,1]$ to $\R$, with the inner product
$$\inn{f,g}\ =\ \int_0^1 f(t)g(t)dt.$$%
Let $f$ be the element of $V$ given by $f(t) = t^2$.  Compute
$\norm{f}$.

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\problemno{8} Let $A = \BRMAT{1&0\cr6&-1}$.  
\par\noindent{\bf(a):\ }
Find a matrix $P$ which diagonalizes $A$.
\par\noindent{\bf(b):\ } 
What is $P^{-1}AP$?

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\problemno{10}
Let $A = \BRMAT{3&0\cr8&-1}$.
\vspace{0.05in}
\par\noindent{\bf(a):}\ find the characteristic equation of $A$
\par\noindent{\bf(b):}\ find the eigenvalues of $A$
\par\noindent{\bf(c):}\ find bases for the eigenspaces of $A$

\vspace{0.1in}

\problemno{11}
Let $A = \BRMAT{0&-1\cr1&0}$.  Find an invertible matrix $P$ and a diagonal
matrix $D$ such that $P^{-1}AP = D$.  [{\sc Note:}\ There will be complex
numbers in your answer.]

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\problemno{12}
Given square matrices $P$, $A$ of the same size, with $P$ invertible,
simplify $(P^{-1}AP)^{10}$.

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\problemno{13}
Let $V$ be an inner product space.  We are given an orthonormal basis
$\setof{a,b,c}$ for $V$, and another element $x$ of $V$, which is a
unit vector.  We know that
$\inn{a+b,x} = 5$, $\inn{c,x} = 4$, and $\norm{a+x} = 4$.

\par\noindent{\bf(a):}\ Compute $\inn{c,b+c+x}$.

\par\noindent{\bf(b):}\ Compute $\inn{a,x}$.

\par\noindent{\bf(c):}\ Express $x$ as a linear combination of $a,b,c$.

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\problemno{14} 
Let $V$ be the vector space of all continuous functions fom $[0,1]$ to $\R$.
Define an inner product on $V$ by
$$\inn{f,g} = \int_0^1 f(t)g(t)dt.$$
Let $h$ be the element of $V$ given by $h(t) = t^3$.
\vspace{0.1in}
\par\noindent{\bf(a):}\ Compute $\norm{h}$.
\par\noindent{\bf(b):}\ Exhibit a basis for the subspace $W$ of $V$ consisting 
of functions having the form $a+bt$, where $a,b \in \R$.
\par\noindent{\bf(c):}\ Exhibit an orthonormal basis for $W$.

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\problemno{15}
{\bf(a):}\ Exhibit three different $2 \times 2$ orthogonal matrices, all 
having determinant $1$.

\par\noindent
{\bf(b):}\ Exhibit two different $3 \times 3$ orthogonal matrices, both
having determinant $1$.

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\problemno{16}
Let $\R^3$ have the standard inner product (dot product).  Let $W$ be the
subspace of $\R^3$ given by 
$$W = \left\{ \BRMAT{t\cr t\cr t}: t \in \R \right\}.$$%

\par\noindent{\bf(a):\ } Exhibit a nonzero element of $W$.
\par\noindent{\bf(b):\ } Find a basis for $W^\perp$.  [{\sc Hint.}\ It
has dimension $2$.]

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\problemno{17} {\bf(a):}\ Let $A,B$ be orthogonal matrices (of the same size).  
Show that $AB$ is orthogonal.
\par\noindent{\bf(b):}\ Show that a $3 \times 3$ real matrix cannot have
eigenvalues $i, i, 1$.

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\problemno{18}
{\bf(a):\ } 
Let $A$ be an $m \times n$ matrix, and let 
$b \in \R^m$.  State precisely what it means for $x \in \R^n$ to be a
{\it least squares solution\/} of the equation $Ax = b$.  (Your answer
should not involve matrix transpose -- that has to do with how you
calculate least squares solutions.  Do not be vague.  A single sentence
will suffice.)
\par\noindent{\bf(b):}\ Explain how to find all least squares solutions to
$Ax = b$.  You may assume that the reader knows how to solve systems of 
linear equations.  Use a complete sentence or sentences.
\par\noindent{\bf(c):}\ Under what circumstances is it the case that 
the solution set of $Ax = b$\\
equals \setofh{least squares solutions to $Ax = b$}?
[Perhaps your answer to (a) will help you do this part.]
\par\noindent{\bf(d):\ } 
Find the least squares solution(s) to $Ax = b$, where
$$A = \BRMAT{1&1\cr-1&1\cr-1&2},\ \ \ b = \BRMAT{7\cr0\cr-7}.$$

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\problemno{19}\ \ Let $\R^3$ have the Euclidean inner 
product.  Let $W$ be the subspace of $\R^3$ given by
$$W\ =\ \left\{ \BRMAT{t\cr-t\cr t}: t \in \R \right\}.$$%
\par\noindent{\bf(a):\ } Find a basis for $W$.
\par\noindent{\bf(b):\ } Find a basis for $W^\perp$.
[{\sc Note:}\ $\dim W^\perp = 2$, so if you find two linearly independent
elements of $W^\perp$ (perhaps by inspection), then they will form a basis for
$W^\perp$.]
\par\noindent{\bf(c):\ } Find an orthonormal basis for $W$.
\par\noindent{\bf(d):\ } Find an orthonormal basis for $W^\perp$.
\par\noindent{\bf(e):\ }
Compute the projection of $\BRMAT{1\cr2\cr3}$ onto $W$.  
\par\noindent{\bf(f):\ }
Compute the projection of $\BRMAT{1\cr2\cr3}$ onto $W^\perp$.  You do not
have to simplify your answer.
\par\noindent [{\sc Hint.}\ \ You can deduce (f) by means of (d) or (e), but 
it's easiest to use (e).]

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\problemno{20}\ \ {\bf(a):\ }
Complete the following definition.  A square matrix $A$ is 
\vspace{0.15in}
\par\noindent{\it diagonalizable\/} if there exists \blank3
\vspace{0.15in}
\par\noindent such that \blank{4.5}.
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\par\noindent{\bf(b):\ } Assume that $A$ is diagonalizable.  Prove that
$A^2$ is diagonalizable.

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\problemno{21}\ \ Let 
$V = \shC[0,1] = \setofh{continuous functions from $[0,1]$ to $\R$}$,
with the inner product given by $\inn{f,g} = \int_0^1 f(t)g(t)dt$.  Find a
positive real number $c$ such that $\norm{ct} = 1$.

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\problemno{22}\ \ Let $A$ be a $3 \times 3$ matrix with 
characteristic polynomial $\lambda^3 - \lambda^2$.  
\vspace{0.1in}
\par\noindent{\bf(a):\ } What are the eigenvalues of $A$?
\par\noindent{\bf(b):\ } For each eigenvalue $\lambda$ of $A$, list all
possible values for the dimension of the corresponding eigenspace $E_\lambda$.
(You do not have to exhibit specific matrices $A$.)
\par\noindent{\bf(c):\ } What can you deduce from (b) about the possible values 
for the nullity of $A$?  Explain briefly.
\par\noindent{\bf(d):\ } Exhibit a $3 \times 3$ diagonal matrix $A$ with
characteristic polynomial $\lambda^3 - \lambda^2$.

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\problemno{23} Let $\R^3$ have the standard inner product.  Let
$$v_1 = \BRMAT{-1\cr0\cr0},\ v_2 = \BRMAT{0\cr3/5\cr4/5},
  \ v_3 = \BRMAT{0\cr4/5\cr-3/5},\ w = \BRMAT{1\cr2\cr3}.$$%
Then \setof{v_1,v_2,v_3}\ is an orthonormal basis for $\R^3$.  Find the
coordinates of $w$ \WRT\ this basis.

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\problemno{24}  Let $P_1$ be the vector space of polynomials of degree
$\leq 1$, with the inner product
$$\inn{p,q}\ =\ \int_0^1 p(x)q(x)dx.$$%
Find an orthonormal basis for $P_1$.

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\problemno{25}  Let $V = \R^2$ with the standard inner product.  Use
pictures to describe all orthonormal bases for $V$.