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{\bf{Homework 5: Linear transformations; matrices. Part 2. \\
Due Monday February 27.}}
\begin{enumerate}
\item Consider the linear space $V$ of degree $n$ polynomials over a field $F\subset \C$
(that is, the space of all functions
$f:F\to F$ of the form $f(x)=a_n x^n+\dots +a_1 x +a_0$, where $a_0, \dots, a_n \in F$).
\begin{enumerate}[(a)]
\item Find the dimension of the space $V$.
\item Let $D:V\to V$ be the linear map $D(f)=f'$ (the derivative).
\item Find the kernel and image of $D$.
\end{enumerate}
\item Let $C(\R)$ be the space of all infinitely differentiable functions $f:\R\to \R$.
Let $D(f)= f''+f $. Show that $D$ is a linear operator on the space $C(R)$, and describe its kernel.
Is its kernel finite-dimensional? Make a guess at its dimension (you do not have to include a rigorous proof, but explain your guess.)
\item Let $V$ be an arbitrary vector space over a field $F$, and let $P:V\to V$ be a linear operator with the property that $P^2=P$ (here by $P^2$ we mean $P$ composed with itself). Such linear operators are called \emph{projectors}.
\begin{enumerate}
\item Prove that $V=\Ker(P)\oplus\im(P)$.
\item Make an example of such a linear operator on $\R^3$.
\end{enumerate}
\item Problem 5.1 from J\"anisch
\item Problem 5.2 from J\"anisch
\item Problem 5.3 from J\"anisch
\item Problem 7.1 from J\"anisch.
In addition, find a basis for the space $\Ker(A)$ for the matrix of this system of equations.
\item Problem 7.2 from J\"anisch.
In addition, find a basis for the space $\Ker(A)$ for the matrix of this system of equations.
\end{enumerate}
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