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\begin{definition*}
    $A \in \mathcal{M}_{m, n}(F)$
    \begin{itemize}
        \item The column rank of $A$, $r(A)$ is
            the dimension of the span of the
            column vectors of $A$ in $F^m$, i.e.
            if $A = (c_1, \dots, c_n)$ then $r(A)
            = \dim_F \Span \{c_1, \dots, c_n\}$.
        \item Similarly, the row rank is the
            column rank of $A^\top$.
    \end{itemize}
\end{definition*}

\begin{remark*}
    If $\alpha$ is a linear map represented by $A$
    with respect to some basis, then
    \[ r(A) = r(\alpha) = \dim \Image \alpha \]
\end{remark*}

\begin{proposition*}
    Two matrices are equivalent if and only if
    $r(A) = r(A')$.
\end{proposition*}

\begin{proof}
    \begin{enumerate}[(1)]
        \item[$\Rightarrow$] If $A$ and $A'$ are equivalent,
            then they
            correspond to the same linear map $\alpha$
            written in two different bases
            \[ r(A) = r(\alpha) = r(A') \]
        \item[$\Leftarrow$] $r(A) = r(A') = r$,
            then both $A$
            and $A'$ are equivalent to:
            \[ \left(
            \begin{tabular}{c|c}
                $I_r$ & $0$ \\
                \hline
                $0$ & $0$
            \end{tabular} \right)  \]
            so $A$ and $A'$ are equivalent.
    \end{enumerate}
\end{proof}

\begin{theorem*}
    $r(A) = r(A^\top)$ (column rank is the same as
    row rank)
\end{theorem*}

\begin{proof}
    Exercise.
\end{proof}

\subsection{Elementary operations and elementary
matrices}

Special case of the change of basis formula. \\
Let $\alpha : V \to W$ be a linear map,
$(\mathcal{B}, \mathcal{B}')$ bases of $V$,
$(\mathcal{C}, \mathcal{C}')$ bases of $W$.
\[ [\alpha]_{\mathcal{B}, \mathcal{C}} \to
[\alpha]_{\mathcal{B}', \mathcal{C}'} \]
If $V = W$, $\alpha : V \to V$ linear then we call
it an endomorphism.
\begin{itemize}
    \item $\mathcal{B} = \mathcal{C}$,
        $\mathcal{B}' = \mathcal{C}'$
    \item $P$ is change of matrix from
        $\mathcal{B}'$ to $\mathcal{B}$.
\end{itemize}
then
\[ [\alpha]_{\mathcal{B}', \mathcal{B}'} = P^{-1}
[\alpha]_{\mathcal{B}, \mathcal{B}} P \]

\begin{definition*}
    $A$, $A'$ are $n \times n$ (square) matrices,
    we say that $A$ and $A'$ are \emph{similar}
    (or \emph{conjugate}) if and only if:
    \begin{itemize}
        \item $A' = P^{-1}AP$
        \item $P$ is $n \times n$ square
            invertible.
    \end{itemize}
\end{definition*}

\noindent
Central concept when we will study diagonalisation
of matrices. (Spectral theory)

\subsection{Elementary operations and elementary
matrices}

\begin{flashcard}
\begin{definition*}
    Elementary \emph{column} operation on an $m
    \times n$ matrix $A$:
    \begin{enumerate}[(i)]
        \item \cloze{swap columns $i$ and $j$ ($i \neq
            j$)}
        \item \cloze{replace column $i$ by $\lambda$
            times column $i$ ($\lambda \neq 0$,
            $\lambda \in F$)}
        \item \cloze{add $\lambda$ times column $i$ to
            column $j$ (with $i \neq j$)}
    \end{enumerate}
\end{definition*}
\end{flashcard}

\begin{itemize}
    \item Elementary row operations: analogous way
    \item Elementary operations are invertible
    \item These operations can be realised through
        the action of \emph{elementary matrices}.
        \begin{enumerate}[(i)]
            \item $i, j$, $i \neq j$.
                \begin{center}
                    \includegraphics[width=0.6\linewidth]
                    {images/7940f95e512b11ed.png}
                \end{center}
            \item $i$
                \begin{center}
                    \includegraphics[width=0.6\linewidth]
                    {images/974e031a512b11ed.png}
                \end{center}
            \item $i, j, \lambda$, $i \neq j$
                \[ C_{i, j, \lambda} = \id + E_{i,
                j} \]
                \begin{center}
                    \includegraphics[width=0.6\linewidth]
                    {images/e69a9d98512b11ed.png}
                \end{center}
        \end{enumerate}
\end{itemize}
Link between elementary operations / matrices: \\
an elementary column (row) operation can be
performed by multiplying $A$ by the corresponding
elementary matrix from the right (left) $\to$
Exercise. \myskip
Now a constructive proof that any $m \times n$
matrix is equivalent to
\[ \left(
\begin{tabular}{c|c}
    $I_r$ & $0$ \\
    \hline
    $0$ & $0$
\end{tabular} \right) \]
\begin{itemize}
    \item Start with $A$. If all entries are zero,
        done.
    \item Pick $a_{ij} = \lambda \neq 0$. Swap
        rows $i$ and 1 and swap columns $j$ and 1.
        Then $\lambda$ is in position (1, 1)
    \item Multiply column 1 by $\frac{1}{\lambda}$
        to get 1 in position (1, 1).
    \item Now clean out row 1 and column 1 using
        elementary operations of type (iii).
    \item Iterate with $\tilde{A}$ (the $(m - 1)
        \times (n - 1)$ sub matrix)
    \item Then at the end of the process we will
        have shown that
        \[ \left(
        \begin{tabular}{c|c}
            $I_r$ & $0$ \\
            \hline
            $0$ & $0$
        \end{tabular}
        \right) \equiv Q^{-1} AP = \ub{E_p' \cdots
        E_1'}_{\text{row operations}} A \ub{E_1
        \cdots E_c}_{\text{column operations}} \]
\end{itemize}

\subsubsection*{Variation}

\begin{itemize}
    \item Gauss' pivot algorithm. If you use
        \emph{only} row operations, we can reach
        the so called ''row echelon form'' of the
        matrix
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/dfa1f3dc512c11ed.png}
        \end{center}
    \item Assume that $a_{i1} \neq 0$ for some $i$
    \item Swap rows $i$ and 1
    \item Divide first row by $\lambda = a_{i1}$,
        to get 1 in (1, 1)
    \item Use 1 to clean the rest of the first
        column
    \item Move to second column
    \item Iterate.
\end{itemize}
This procedure is exactly what you do when solving
a linear system of equations: Gauss' pivot
algorithm

\subsubsection*{Representation of square
invertible matrices}

\begin{flashcard}[I-n-row-operations-only]
\begin{lemma*}
    If $A$ is $n \times n$ \emph{square
    invertible} matrix, then \cloze{we can obtain $I_n$
    using row elementary operations only (or
    column operations only).}
\end{lemma*}
\end{flashcard}

\begin{proof}
    \begin{itemize}
        \item We do the proof for column
            operations. We argue by induction on
            the number of rows
        \item Suppose that we could reach a form
            where the upper left corner is $I_k$.
            We want to obtain the same structure
            with $k \to k + 1$.
        \item Claim: there exists $j > k$ such
            that $\lambda = a_{k + 1, j} \neq 0$.
            Otherwise the vector $\delta_{i(k +
            1)}$ is \emph{not} in the span of the
            column vectors of $A$ (exercise) which
            contradicts the assumption that $A$ is
            invertible.
        \item Swap column $k + 1$ and $j$
        \item Divide column $k + 1$ by $\lambda
            = a_{k + 1, j} \neq 0$
        \item Use 1 to clear the rest of the $k +
            1$-th row using elementary operation
            of type (iii).
        \item This completes the inductive step.
        \item Continue until $k = n$.
    \end{itemize}
\end{proof}

\myskip
Outcome:
\[ AE_1 \cdots E_c = I_n \]
\[ \implies A^{-1} = E_1 \cdots E_c \]
so this gives an algorithm for computing $A^{-1}$.
(useful for solving $AX = F$, linear system of
equations).

\begin{proposition*}
    Any invertible square matrix is a product of
    elementary matrices.
\end{proposition*}
