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\title{Math 23a Problem Set 5}
\dedicatory{due October 27th}

\begin{document}\maketitle
 
Let $f\colon V\times V \to \bR$ be a bilinear form. We say that $f$ is
symmetric if $f(v_1,v_2)=f(v_2,v_1)\forall v_1,v_2 \in V$. We say that
$f$ is anti-symmetric if $f(v_1,v_2)=-f(v_2,v_1)\forall v_1,v_2 \in V$.

\begin{enumerate}\item Let $V$ be a two-dimensional vector space, and $\set{e_1,e_2}$
and $\set{v_1,v_2}$ be two bases of $V$, such that $v_1=e_1
+e_2,v_2=e_1 -e_2$ and $f\colon V\times V \to \bR$ be the bilinear form
such that $f(v_1,v_1)=f(v_2,v_1)=f(v_2,v_2)=1$ and $f(v_1,v_2)=0$.

\begin{enumerate}\item Find $f(e_1,e_2)$.

\item Show that there exists unique pair $f_s,f_{as}$ of bilinear forms on
$V$ such that $f_s$ is symmetric, $f_{an}$ is anti-symmetric and
$f=f_s +f_{an}$.

\item Find $f_s (e_1,e_2)$.

\end{enumerate}\item Let $f$ be a bilinear form on a vector space $V$ and $B=\{v_1,\dots,v_d\}\in
V$ be a basis of $V$. Consider the $d\times d$ matrix $M^B (f)$ such that
$M^B(f)_{i,j}=f(v_i,v_j),1\leq i,j\leq d$.

\begin{enumerate}\item Show that the bilinear form $f$ is symmetric if and only if
$M^B (f)_{i,j}=M^B (f)_{j,i}\forall i,j,1\leq i,j\leq d$ for some basis
$B$ in $V$. Show also that in this case
$M^B (f)_{i,j}=M^B (f)_{j,i}\forall i,j,1\leq i,j\leq d$ for every
basis $B$ in $V$. 


\item Show that the bilinear form $f$ is antisymmetric if and only if
$M^B (f)_{i,j}=-M^B (f)_{j,i}\forall i,j,1\leq i,j\leq d$ for some basis
$B$ in $V$.

\item Let $V=\bR ^3,f$ be a bilinear form on $V$ such that
$f(e_i,e_j)=\delta ^i_j$ where $\delta ^i_j =1$ if $i=j$ and $0$
otherwise.
Let $v_1=e_1+e_2+e_3,v_2=e_1-e_2+e_3,v_3=e_1+e_2-e_3$. Show that 
 $B=\{v_1,v_2,v_3\}$ is a basis of $V$ and write down the matrix $M^B
(f)$. 

\end{enumerate}\item Let $P_3$ be the vector space of polynomials of degree $\leq 3$ and $(
,)\colon P_3 \times P_3 \to \bR$ be the bilinear form given by
$(p(x),q(x)) =\int_{x=-1}^{x=1}p(x)q(x)dx$. 

\begin{enumerate}\item Show that $P_3,(,)$ is an Euclidean space.

Let $B=\{v_0,v_1,v_2,v_3\}$ be a basis in $P_3$ where $v_0 =1,v_1 =x
,v_2 =x^2,v_3 =x^3$ . 

\item Find real numbers $a,b,c,d,e,f$ such that vectors 
$w_0=v_0,w_1=v_1 +av_0,w_2 =v_2 +bv_1 +cv_0,w_3 =v_3 +dv_2+ ev_1
+fv_0$ form an orthogonal basis of $P_3$.

\item Find positive real numbers $r_i,0\leq i \leq 3$ such that vectors 
$e_i =r_i w_i,0\leq i \leq 3$ form an orthonormal basis of $P_3$.

\item EXTRA CREDIT. Replace $P_3$ by $P_n$ and show that $v_n=n!/(2n)!
(d/dx)^n (x^2-1)^n$

\end{enumerate}
\item
\begin{enumerate}Let $e_1,\dots,e_d$ and $v_1,\dots,v_d$ be two bases of a vector space
$V$. Since $e_1,\dots,e_d$ is a basis for any $j,1\leq j\leq d$ we can
find numbers $a^{i}_j,1\leq i\leq d$ such that $v_j=\sum_{1\leq i\leq
d}a^{i}_j e_i$. Prove that the $d\times d$ matrix $A=\{a^{i}_j \}$ is
invertible.

\item Let $f$ be a bilinear form on $V$ such that $f(e_i,e_j)=\delta_{i,
j},1\leq i,j\leq d$ where $\delta_{i,
j} =1$ if $i=j$ and is equal to $0$ otherwise. Find $f(v_i,v_j)$. 

\end{enumerate}\end{enumerate}\end{document}
