\documentclass{standalone}
\usepackage{tikz, amsmath}
\usetikzlibrary{arrows.meta}

\DeclareMathOperator{\im}{im}

\begin{document}
\begin{tikzpicture}
[
map/.style={->, shorten <=7pt, shorten >=7pt, -{>[scale=1.5]}},
index/.style={color=black!50, dashed, shorten <=2pt, shorten >=2pt},
]

\node (V)     at (2, 6) {\( V \)};
\node (kerT)  at (2, 2) {\( \ker T \)};
\node (zero1) at (2, 0) {\( 0 \)};

\node (dim)   at (0, 3) {\( \dim V \)};
\node (null)  at (1, 1) {\( \dim\ker T \)};

\node (W)     at (6, 8) {\( W \)};
\node (imT)   at (6, 4) {\( \im T \)};
\node (zero2) at (6, 0) {\( 0 \)};

\node (rank)  at (7, 2) {\( \dim\im T \)};

\draw (V)     -- (kerT);
\draw (kerT)  -- (zero1);

\draw (W)     -- (imT);
\draw (imT)   -- (zero2);

\draw (V)     -- (imT)   [map];
\draw (kerT)  -- (zero2) [map];
\draw (zero1) -- (zero2) [map];

\draw [index] (V)    .. controls (0, 6) .. (dim);
\draw [index] (dim)  .. controls (0, 0) .. (zero1);
\draw [index] (kerT) .. controls (1, 2) .. (null);
\draw [index] (null) .. controls (1, 0) .. (zero1);
\draw [index] (imT)  .. controls (7, 4) .. (rank);
\draw [index] (rank) .. controls (7, 0) .. (zero2);
\end{tikzpicture}
\end{document}