
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 6}

\usepackage{amssymb} 

\usepackage{tikz-cd}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 26 March. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

Let $R$ be a commutative ring with $1$. Recall that if $M$ and $N$ are $R$-modules then there
  exist an $R$-module $M \otimes_R N$ and an $R$-bilinear map $\otimes: M \times N \rightarrow M \otimes N$
  with the following {\em universal property}: for every bilinear $\psi: M \times N \rightarrow L$ there is
  a unique linear $\rho: M \otimes_R N \rightarrow L$ such that $\psi = \rho \circ \otimes$.
  It is not hard to see \footnote{$M \otimes_R N$ is the quotient of the free module $F$ on $M \times N$ by the
    least submodule $X$ such that $(m, n) \mapsto \otimes (m, n) + X$ is a bilinear map from $M \times N$ to $F/X$,
    and $m \otimes n = (m, n) + X$. Elements of $F$ have the form $\sum_{i=1}^n r_i(m_i, n_i)$,
    so elements of $F/X$ have the form $\sum_{i=1}^n r_i(m_i, n_i) + X$, and the bilinearity of $\otimes$  
    implies that $\sum_{i=1}^n r_i(m_i, n_i) + X = \sum_{i=1}^n (r_i m_i) \otimes n_i$.} that every element
  of $M \otimes_R N$ can be written as $\sum_{i=1}^n m_i \otimes n_i$. This type of expression is typically
  {\em not unique}.

\medskip

\begin{enumerate}

\item  Let $R$ and $S$ be commutative rings with $1$ such that $R$ is a subring of $S$ (we recall that
  by convention this implies $1_R = 1_S$).

  \begin{enumerate}

  \item  Prove that $S$ is an $R$-module if we define addition to be $+_S$, and scalar multiplication 
  to be $\times_S \restriction R \times S$. 

  \item  Let $M$ be an $R$-module and let $N = S \otimes_R M$. Prove that for each $t \in S$  there is a unique  
  $R$-linear map $\alpha_t : N \rightarrow N$ such that $\alpha_t(s \otimes m) = (t s) \otimes m$ for all $s$ and $m$. 

  \item  Prove that if we define a scalar multiplication from $S \times N$ to $N$ by $t x = \alpha_t(x)$, then
    $N$ becomes an $S$-module.

  \end{enumerate}

\item  Prove that if $f \in {\mathbb R}[x]$ and $\alpha \in {\mathbb C}$ with $f(\alpha) = 0$, then
  $f(\bar\alpha) = 0$. Use this plus the Fundamental Theorem of Algebra to prove that every irreducible
  monic polynomial in ${\mathbb R}[x]$ is of one of the two forms:

  \begin{itemize}

  \item  $x - \lambda$ for $\lambda \in {\mathbb R}$. 
    
  \item  $x^2 - 2 \lambda x + (\lambda^2 + \mu^2)$ for $\lambda, \mu \in {\mathbb R}$ with $\mu \neq 0$.

  \end{itemize}

\item  Let $V$ be a real vector space of dimension two and let $T: V \rightarrow V$ be linear. Prove that there
 is a basis for $V$ such that the matrix of $T$ has one of the following three forms:     
 \begin{itemize}

 \item $\left ( \begin{array}{rr} \lambda & 0 \\ 0 & \mu \\ \end{array} \right )$ for $\lambda, \mu \in {\mathbb R}$.
 
\item $\left ( \begin{array}{rr} \lambda & 0 \\ 1 & \lambda \\ \end{array} \right )$ for $\lambda \in {\mathbb R}$.

\item
  $\left ( \begin{array}{rr} \lambda & -\mu \\ \mu    & \lambda \\ \end{array} \right )$ for $\lambda, \mu \in {\mathbb R}$
   with $\mu \neq 0$. 
  
 \end{itemize}

    Hint: Imitate the proof for the existence of Jordan normal form. 

\item   Let $V$ be a real vector space of dimension two and let $E$ be a subspace of $Lin(V, V)$ such that 
  \begin{itemize}
  \item $\dim(E) = 2$.
  \item The identity map $I$ is in $E$. 
  \item Every nonzero element of $E$ is invertible.
  \end{itemize}

  Prove that there is a basis for $V$ such that the set of matrices of elements of $E$ is precisely the set of matrices of the
  form  $\left ( \begin{array}{rr} \lambda & -\mu \\ \mu    & \lambda \\ \end{array} \right )$ for $\lambda, \mu \in {\mathbb R}$. 


\item   Recall that if $V$ is the real vector space  of continuous  functions from $[0, 1]$ to $\mathbb R$ then
  defining $f \cdot g = \int_0^1 f(x) g(x) dx$ gives an inner product. Let $W$ be the subspace consisting of polynomials
  of degree at most $3$.

  \begin{enumerate}

  \item Find an orthonormal basis for $W$.

  \item Prove that every $f \in V$ can be written uniquely as $f_0 + f_1$ where $f_0 \in W$ and $f_1 \in W^\perp$.

  \end{enumerate}   

\item Recall that a symmetric real matrix $A$ is {\em positive semidefinite} if and only if $x^T A x \ge 0$ for all
  $x \in {\mathbb R}^n$, equivalently $A$ has no negative  eigenvalues.

  Let $V$ be a real inner product space, let $v_i \in V$ for $1 \le i \le n$ and let $A$ be the
  matrix whose $(i, j)$ entry is $v_i \cdot v_j$.  Prove that $A$ is  positive semidefinite.
  Conversely, prove that every positive semidefinite matrix arises in this way for a suitable
  choice of $V$ and $v_i$. Hint: Spectral theorem for real symmetric matrices. 


  
   
\end{enumerate}  






\end{document} 
