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{\bf{Homework 4: Linear transformations, Part 1. \\}}
\begin{enumerate}
\item Let $V$ and $W$ be vector spaces over a field $F$.
Let $A:V\to W$ be a linear transformation that has an inverse function $B:W\to V$.
Prove that $B$ has to be a linear transformation.
\item Problem 4.1 from J\"anisch
\item Think of $\C$ as a $2$-dimensional vector space $V$ over $\R$, and
let $A:V\to V$ be the linear transformation of $V$ given by the multiplication by $1+2i$ in $\C$.
Write the matrix of $A$ with respect to the standard basis of $V$.
%\item Problems (2) and (4) from 5.4 ``Test'' on p. 93 of J\"anisch.
\item Let $V=\R^n$. Prove that every linear functional $f:V\to \R$ is of the form
$f(x_1, \dots, x_n)= a_1 x_1 +\dots +a_n x_n$ for some constants $a_1, \dots, a_n \in \R$.
{\it {(Hint: think of what it does to the standard basis vectors).} }
{\bf Remark:} This shows that, in fact, the space $V^\ast :=\mathrm{Hom}_\R (V, \R)$ of all linear functionals on $V$ is isomorphic to $V$ (for every finite-dimensional space $V$). For infinite-dimensional spaces this is, generally, not true.
\item Consider the linear space $V$ polynomials of degree not greater than $n$ 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 $a_0, \dots, a_n \in F$).
Let $D:V\to V$ be the linear map $D(f)=f'$ (the derivative). Find the kernel and image of $D$.
\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 $V=\ker(P)\oplus {\mathrm {Im}}(P)$.
\item Make an example of such a linear operator on $\R^3$.
\end{enumerate}
\end{enumerate}
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