숙제3

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C. Linear Transformations and Their Matrices

\begin{enumerate}
\item
If $A$ and $B$ are linear transformations (on the same vector space),
then a necessary and sufficient condition that both $A$ and $B$ be invertible is that
both $AB$ and $BA$ be invertible.
\item
If $A$ and $B$ are linear transformations on a finite-dimensional vector space,
and if $AB=I$, then both $A$ and $B$ are invertible.
\item
If $A$ and $B$ are linear transformations (on the same vector space) and if
$AB=I$, then $A$ is called a \textem{left inverse} of $B$ and
$B$ is called a \textem{right inverse} of $A$.
Prove that if $A$ has exactly one right inverse, say $B$, then $A$ is invertible.
(Hint: consider $BA+B-I$.)
\item
Prove that if $e_1$, $e_2$, and $e_3$ are the complex matrices
$$
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},
\quad
\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix},
\quad
\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}
$$
respectively (where $i=\sqrt{-1}$), then
$e_1^2=e_2^2=e_3^2=-I$,
$e_1e_2=-e_2e_1=e_3$,
$e_2e_3=-e_3e_2=e_1$,
and
$e_3e_1=-e_1e_3=e_2$.
\item
{교과서 19쪽 Exercise 3}.
\item
Let $\mathcal{L}$ be the space of linear maps from $U^n$ to $V^m$.
And let $\mathcal{M}$ be the space of $m\times n$ matrices.
Choose and fix a basis for $U$ and a basis for $V$.
With respect to these bases, to each linear map $T\in \mathcal{L}$
can we associate an $m\times n$ matrix $A_T$ (as was explained in the spring semester).
Let us denote this association by $\Phi$.
Show that this map $\Phi$ is a vector space isomorphism which also satisfies
the following identity.
$$
\Phi(S\circ T) = \Phi(S)\Phi(T)\quad\text{or}\quad
A_{S\circ T}=A_S A_T
$$
and
$$
\Phi(T^{-1}) = \Phi(T)^{-1}\quad \text{or} \quad
A_{T^{-1}} = A_T^{-1}
$$

\end{enumerate}