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\begin{document}
If $V$ is a vector space  $B=\{e_1,...,e_d\}$ be a basis of $V$
and $T:V\ra V$ is a linear transformation we denote by $A_B (T)$ the
matrix of $T$ in the basis $B$. That is 
$ A_B (T)=(a^i_j),1\leq i,j\leq d$ is  a matrix such that 
$T(e_j)=\sum _{i=1}^d a^i_j e_i$.

1. a)Let  $B_0=:\{e_1,e_2\}$ be the standard basis of $ \bR ^2 ,T:\bR
   ^2\ra \bR ^2$ be a linear transformation such that 
 $$A_{B_0} (T)=\begin{pmatrix}2&1\\1&1\quad
\end{pmatrix}$$  [that is  
$$T(e_1) =2e_1 +e_2 ,T(e_2) =e_1 +e_2 ]$$

Find a basis $B=:\{f_1,f_2\}$  of $ \bR ^2$ such that the matrix  $A_B
(T)$ is diagonal. That is  $A_B (T)=\Lambda =\begin{pmatrix}\lambda
_1&0\\0&\lambda _2\quad 
\end{pmatrix}$ 

b)find the numbers $\lambda _1$ and $\lambda _2$.

c)Let $f_1=c^1_1 e_1+c^2_1 e_2,f_2=c^1_2 e_1+c^2_2 e_2, c^i_j \in \bR$.
 
Show that $A=C\Lambda C^{-1}$ where $A:=A_{B_0} (T)$.

$C=\begin{pmatrix}c^1_1
&c^2_1\\c^1_2&c^2_2\quad 
\end{pmatrix}$ 

d)Let $a=(3-\sqrt 5)/2$. Find the elements of the matrix
$a^{1000}A^{1000}$ with the precision of 10 percent.  "With the precision of 10
percent" means that if $e$ is
the exact (unknown) answer and $a$ is the approximate answer which you give, you
must have $|a-e|/a < 0.1$.
\vskip 0.2cm
2.a) Let $f$ be a differentiable function on $\bR$ such that $f'(x)\equiv
1$.

Prove that there exists $c\in \bR$ such that $f(x)=x+c$ for all $x\in
\bR$.

b)Find all twice differetiable functions $F$ on $\bR$ such that $F''(x)\equiv
1$.
\vskip 0.2cm

Let $f:Mat(n,n)\ra Mat(n,n)$ be a  function such that $f(A)=A^2$. 
3. a)Show that the function $f$ is differentiable and for any $n\times n$
matrix $H$ find the differential $df(H):Mat(n,n)\ra Mat(n,n)$. 

 We can identify the vector space $Mat(n,n)$ with $\bR ^{n^2}, A\ra
\{a_{i,j} \}$. Therefore we can define the partial derivatives $\partial
f/\partial a_{i,j}$.

b) Find   the partial derivatives $\partial
f/\partial a_{i,j}$.

c) Find the differential $dF(H):Mat(n,n)\ra Mat(n,n)$ where $F(A):=A^3$.
\vskip 0.2cm

4. a) Let $f$ be a differentiable function on $\bR$ and $F$ be and
function on $\bR ^2$ defined by $F(x,y):=x^7 f(y/x^2)$.

Show that $x\partial F/\partial x +2y\partial F/\partial y=7F$

b) Find a generalization of this result.
\vskip 0.2cm
If $f$ is an infinitely differetiable function on $\bR$ at $a\in \bR$
we can define the Taylor series 
$$P_f(a,x):=\sum _{n\geq 0}c_n (x-a)^n,c_n :=f^{(n)}(a)/n!$$.
 We define by $I_a \subset [0,\infty)$ as the set of numbers $r\geq 0$
 such that  $c_n r^n \ra 0$ for  $ n\ra \infty $ and write
$r_f(a):=lub (I_a)$. [It could be that $r_f(a)=\infty$]. 

 Let $f$ be a function on $\bR -\{0,1,2\}$ given by 
$f(x):=\frac {1}{x(x-1)(x-2)}$.

5. a)Show that  $f$ is an infinitely differetiable function on $\bR
-\{0,1,2\} $

b)Find $r_f(a)$  for any $a\in \bR ,a\neq 0,1,2$
\vskip 0.2cm
Let $V$ be a vector space, $\mathcal Q (V)$ the space of quadratic
functions on $V$ and $\mathcal B^2_{s}(V)$ be the space of symmetric
biliner forms on $V$. To any  symmetric biliner form $f$ on $V$ we
associate a function $q(f)$ on $V$ where $q(f)(v):=f(v,v)$.

6. a) Show that for any $f \in \mathcal B^2_{s}(V)$ the function  $q(f)$
on $V$ is quadratic.

Part a) shows that we have defined a map $\alpha :\mathcal B^2_{s}(V)
\ra \mathcal Q (V),\alpha (f):=q(f)$.

b)Construct explicitely a linear map $\beta :\mathcal Q (V) \ra \mathcal
B^2_{s}(V)$  which is the inverse of $\alpha$.  (Note that it follows from this
that $\alpha$ gives an isomorphism between the vector spaces $\mathcal B^2_{s}(V)$
and $\mathcal Q (V)$.)

c)Suggest  a generalization of of the construction $f \ra q(f)$ and try
to generalize the problems a)-b).
\vskip 0.2cm

7. Let $f$ be a twice differentiable function on $\bR ^2$ and $g: \bR ^2-
   \{x,0\}    \ra \bR ^2$ be given by $g(x,y):=(xy,x/y)$.

Find first and second partial derivatives of the function $F:= f\circ g$
in terms of the partial derivatives of the function $f$

\end{document}
