\documentclass[reqno,12pt]{amsart}
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\DeclareMathOperator{\image}{Im}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\bR}{{\mathbb R}}
\title{Math 23a Final}

\begin{document}
\maketitle

1. (10 pts) Let $V$ be a finite-dimensional vector space,$V',V''\subset V$ linear
subspaces such that $V'\cap V''=\{0\}$ and
$dim(V)=dim(V')+dim(V'')$. Prove that for any $v\in V$ there exist 
$v'\in 'V$ and $v''\in V''$ such that $v=v'+v''$. Show also that for any
$v\in V$ such a pair $(v',v'')$ is unique.

2.a) (10 pts) Let $X\subset \bR$ be a non-empty subset which is similtoneously open
and closed. Show that $X=\bR$.

[A hint. Since $X$ is non empty you can fix a point $x_0\in X$. Look for
the "smallest" point $y\in \bR -X$ which is bigger then $x_0$ and show
that an existence of $y$ would lead to a contradiction]

b) (5 pts) Let $X\subset \bR ^n$ be a non-empty subset which is similtoneously open
and closed. Show that $X=\bR ^n$.

3)a) (10 pts) Let $V$ be a vector space, $A:V\to V$ a linear operator $v_1,v_2 \in
V$ non-zero vectors such that $Av_1=c_1v_1,Av_2=c_2v_2$ where $c_1\neq
c_2 \in \bR$. Prove that the vectors $v_1,v_2$ are linearly independent.

b) (10 pts) Let $v_1,\dots,v_n$ be non-zero vectors in $V$  such that
$Av_1=c_1v_1,\dots,$
$Av_n=c_nv_n$ where $c_1,\dots,c_n \in \bR$ are such that $c_i\neq c_j$
for $i\neq j$. Prove that the vectors $v_1,\dots,v_n$ are linearly independent.

4) (5 pts) Let $f(x,y):\bR ^2\to \bR$ be a twice differentiable function with
continuous second derivatives,$g(h)=f(h,h^3)$. Prove that
$g''(0)=f_{xx}(0,0)$.

5)Let $f(x,y,z):\bR ^3\to \bR$ be  a continuously differentiable function
 such that $f(0,0,0)=0$ and  $f_z(0,0,0)\neq 0$.

a) (5 pts) Show that there exists an open neighboorhood $U\subset \bR ^2$ of the
point $(0,0)$ and a continuously differentiable function  $g(x,y)$ on
$U$ such that $f(x,y,g(x,y))=0$.

b) (5 pts) Assume that $g(0,0)\leq g(x,y)$ for all  $(x,y)\in U$. Show that 
$f_x (0,0,0)=f_y (0,0,0)=0.$
   
6)Let $P_n(x)$ be the Taylor polynomial of degree $n$ for $f(x)=\sin (x)$
at $x_0=1$.

a) (5 pts) Find $P_3(x)$.

b) (10 pts) Let $r_n(x)=\sin(x)-P_n(x)$. Prove that \( \lim_{n\to \infty }r_n(10)=0\)
\end{document}
