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\title{Math 2135: Linear Algebra}
\author{Assignment 3}
\date{due 9 February 2022, end of day}

\begin{document}
\maketitle

Homework should be submitted as a single PDF attachment to \texttt{theojf@dal.ca}. Please be sure to include your surname in the file name.

You are encouraged to work with your classmates, but your writing should be your own. If you do work with other people, please indicate (by name) whom you worked with. You should attempt every question, but it is not expected that you will solve all of them.


\begin{enumerate}
  \item There is an interesting function $\bC \to \bC$ called \emph{complex conjugation} and denoted $z \mapsto \bar z$. It takes a complex number $z = a+b\sqrt{-1}$, and $a,b \in\bR$, to the \emph{conjugate} number $\bar z = a - b\sqrt{-1}$.
  \begin{enumerate}
    \item Think of $V= \bC$ as a vector space over $\bF = \bR$. Is complex conjugation a linear map?
    \item Think of $V = \bC$ as a vector space over $\bF =\bC$. Is complex conjugation a linear map?
  \end{enumerate}
  \item Given any $b,c \in \bR$, define a function $T : \bR^3 \to \bR^2$ by 
  $$ T(x,y,z) = (2x-4y+3z+b, 6x+cxyz).$$
  Show that $T$ is linear if and only if $b=c=0$.
  \item Let $V\subset \bR^\bR$ be the vector space of all differentiable functions. Given $f \in V$, denote its derivative by $f'$.
  \begin{enumerate}
    \item Is the function $V \to \bR^2$ that sends $f \mapsto \bigl(f(1) + f'(2), \int_0^3 f(x)\,\mathrm{d}x\bigr)$ linear? Why or why not?
    \item Is the function $V \to \bR$ that sends $f \mapsto \int_0^3 x^2 f(x)\,\mathrm{d}x$ linear? Why or why not?
    \item Is the function $V \to \bR$ that sends $f \mapsto f'(2)^2$ linear? Why or why not?
  \end{enumerate}
  \item Suppose $T : \bF^4 \to \bF^2$ is a linear map such that $$\ker(T) = \{(x_1,x_2,x_3,x_4) \st x_1=5x_2 \text{ and } x_3=x_1+x_4\}.$$ Show that $T$ is surjective. 
  
  \textbf{Note:} As in lecture, $\ker(-)$ and $\operatorname{null}(-)$ mean the same thing. Similarly, $\im(-)$ and $\operatorname{range}(-)$ mean the same thing.
  \item \begin{enumerate}
   \item Find a linear map $T : \bF^2\to \bF^2$ such that $\ker(T) = \im(T)$ or show that one does not exist.
   \item Find a linear map $T : \bF^3\to \bF^3$ such that $\ker(T) = \im(T)$ or show that one does not exist.
   \end{enumerate}
  \item Let $S : U \to V$ and $T : V \to W$ be linear maps. Show that if any two of the three maps $S$, $T$, and their composition $TS : U \to W$, are invertible, then so is the third one.
  
  \textbf{Remark:} This result is called the \emph{two-out-of-three theorem for invertibility}. 
\end{enumerate}



\end{document}  