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\begin{document}
\begin{center}
{\Large\textbf{Math 2135 - Assignment 10}}\\
\medskip
Due November 9, 2018\\
\end{center}
\bigskip
\thispagestyle{empty}

\begin{enumerate}
\item
 Let $V := \R^+$ be the set of positive real numbers.
\begin{enumerate}
\item
 Is this a vector space over the field $\R$ under the usual operations $+$ and scalar multiples?
\item
 Show that $V$ is a vector space over $\R$ with operations
\begin{align*}
 u \oplus v := u\cdot v \text{ for addition}, \\
 c \odot v := v^c  \text{ for scalar multiples},
\end{align*}
 with $u,v\in V, c\in\R$. Check all properties!
\end{enumerate}
 
\item
 Prove or give a counter example:
\begin{enumerate}
\item
 The union $U_1 \cup U_2$ of two subspaces $U_1,U_2$ of a vector space $V$ is again a subspace of $V$.
\item
 For every subspace $U$ of an $n$-dimensional vector space $V$ there exists $0\leq k\leq n$ and vectors
 $u_1,\dots,u_k\in V$ such that $U = \Span( u_1,\dots,u_k)$.  
\end{enumerate}

  
\item
 Extend the following list of vectors to a basis of the corresponding vector space if possible:
\begin{enumerate}
\item
 $(\vd{1}{-3}{-2},\vd{-2}{6}{4})$ in $\R^3$ 
\item
 $(2+x, 3+2x+x^2)$ in $P_2$.
\end{enumerate}

\item
 Let $A\in F^{m\times n}$. 
 Assume that $f\colon F^n\to F^m,\ x\mapsto A\cdot x$, is bijective. Show that $A$ is invertible.
 What is $f^{-1}$?
 
 Hint: Use the Invertible Matrix Theorem.


\item
\begin{enumerate}
\item
 Let $U=\R^\R$ be the set of all functions from $\R$ to $\R$. Show that $e\colon U\to\R,\ f\mapsto f(5)$, is linear.
\item  
 Let $V$ be the set of real-valued functions that can be integrated over the interval $[0,1]$. Show that
\[ i\colon V\to \R,\ f\mapsto \int_0^1 f(x)\, dx, \]
 is linear. 
\end{enumerate} 

\item
 Show that the following maps are not linear by giving concrete vectors for which the defining properties
 of linear maps are not satisfied.
\begin{enumerate}
%\item
% $f: \R^2\to\R^2, \left[\begin{matrix} x \\ y \end{matrix}\right] \mapsto \left[\begin{matrix} x+1 \\ y+3 \end{matrix}\right]$
\item 
 $g: \R^2\to\R^2, \left[\begin{matrix} x \\ y \end{matrix}\right] \mapsto \left[\begin{matrix} xy \\ y \end{matrix}\right]$
\item 
 $h: \R^2\to\R^2, \left[\begin{matrix} x \\ y \end{matrix}\right] \mapsto \left[\begin{matrix} |x|+|y| \\ 2x \end{matrix}\right]$
\end{enumerate}

\item
 Give the standard matrices for the following linear transformations:
\begin{enumerate}
\item $f:\R^2\to\R^3, \left[\begin{matrix} x \\ y \end{matrix}\right]\mapsto \left[\begin{matrix} 2x+y \\ x \\ -x+y \end{matrix}\right]$;
\item the function $g$ on $\R^2$ that scales all vectors to half their length;
\item the projection $h$ of vectors in $\R^2$ onto the vector $\vt{0}{1}$.   
\end{enumerate}
  
\item
 Let $f\colon\R^2\to\R^3$ be a linear map such that 
\[ f(\left[\begin{matrix} 1 \\ 2 \end{matrix}\right]) = \left[\begin{matrix} 2 \\ 0 \\ -3 \end{matrix}\right], f(\left[\begin{matrix} 3 \\ 2 \end{matrix}\right]) = \left[\begin{matrix} -2 \\ 2 \\ 1 \end{matrix}\right] \]
\begin{enumerate}
\item
 Use the linearity of $f$ to find $f(\left[\begin{matrix} 1 \\ 0 \end{matrix}\right])$ and $f(\left[\begin{matrix} 0 \\ 1 \end{matrix}\right])$.
\item
 Determine the standard matrix of $f$ and $f(\left[\begin{matrix} x \\ y \end{matrix}\right])$ for arbitrary $x,y\in\R$.
\end{enumerate}


\end{enumerate}

\end{document}



