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\title{Math 23a 2-d Midterm exam,Nov.16}

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1. a) Let $G$ be a subset of $\bR^n$.  Say what it means for $G$ to be
closed in terms of convergent sequences.

b) Show that $G$ is closed if and only if $G^c = \bR^n \setminus G$ is
open.

c) Let $f:\bR ^n\ra \bR$ be a function.  For any subset $V\subset \bR$ we
define the preimage $f^{-1}(V)\subset \bR ^n$ by  $f^{-1}(V)=\{x\in \bR^n
| f(x)\in V\}$.  Prove that $f$ is 
continuous if and only if for any open subset $V$ of $\bR$ the preimage
$F^{-1}(V)$ is an open subset of $\bR ^n$.  (note: by part (b) this is
equivalent to showing that $f^{-1}(G)$ is closed for any closed set $G$ if
and only if $f$ is continuous so you can show that instead if you so
desire)

2.  Let $f:\bR \ra \bR^2$ be defined by $f(t) = (e^t, e^{-t})$ and
$g:\bR^2 \ra \bR^3$ be defined by 
$$g(x_1, x_2) = (\frac{x+y}{2},\frac{x-y}{2},xy).$$

Find $(g \circ f)'$ at $t = 0$ first by composing the functions and
evaluating the derivative and then by the chain rule, verifying that the
chain rule gives the same answer.


3.Let $M_n$ be the vector space of $n\times n$ matricies and $F:M_n \ra
  M_n$ be the map given by $F(A)=AA^T$ where $A^T$ is the   traspose of
  $A$.
a)Prove that $F$ is contineous.

b)Prove that $F$ is differentiable and find $D(F)(A)$ for $A\in M_n$.

4.Let $\mathcal B _{an}(\bR ^n)$ be the space of antisymmetric bilinear
  forms on $\bR ^n$.
Find the dimension of  $\mathcal B _{an}(\bR ^n)$ for

a)n=1

b)n=2

c)n=3

d)n=10

5.Let $U\subset \bR ^n$ be an open set ,$F:U\ra \bR ^n$ a continuosly
  differentiable function such that for any $u\in U$ the linear map
  $D(F)(u):\bR ^n \ra \bR ^n$ is an isomorphism. Fix $v\in \bR ^n$ and
  consider the function $f$ on $U$ given by $f(u)=|f(u)-v|^2$.

a)Show that the function $f$ is differentiable 

b)Prove that $D(f)(u)=0$ if and only if $f(u)=v$

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