Tabular custom rules across certain columns

Question

I'm using two custom table rules to reproduce the following table:

The codes below have problems at the edges, how to avoid the rules from producing extra spaces and extra length.

\documentclass{article}
\usepackage{array}
\usepackage{multirow}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{fourier}
\usepackage{setspace}
\setstretch{1.5}
\DeclareMathOperator{\im}{im}

\makeatletter
\newcommand{\Lmidrule}{%
    \noalign {\ifnum 0=`}\fi \hrule height .55pt
    \futurelet \reserved@a \@xhline
}
\newcolumntype{"}{@{\hskip\tabcolsep\vrule width .65pt\hskip\tabcolsep}}
\makeatother

\makeatletter
\def\Cline#1#2{\@Cline#1#2\@nil}
\def\@Cline#1-#2#3\@nil{%
  \omit
  \@multicnt#1%
  \advance\@multispan\m@ne
  \ifnum\@multicnt=\@ne\@firstofone{&\omit}\fi
  \@multicnt#2%
  \advance\@multicnt-#1%
  \advance\@multispan\@ne
  \leaders\hrule\@height#3\hfill
  \cr}
\makeatother

\begin{document}

\begin{center}
\begin{tabular}{c"c"c"c"c"c"c"c"c}\toprule 
& $V_1$ & $V_2$ & $\cdots$ & $V_k$ & $V_{k+1}$ & $\cdots$ & $V_m$ 
\\\Lmidrule
& $\alpha_1$ & $\alpha_2$ & $\cdots$ & $\alpha_k$ &&&& 
\\\Cline{1-5}{.55pt}
\multirow{4}{*}{$\im T$} & $T\alpha_1$ & $T\alpha_2$ & $\cdots$ & $T\alpha_k$ &&&& 
\\
& $\vdots$ & $\vdots$ & & $\vdots$ &&&& $V/\ker T$
\\
& $T^{l_1-2}\alpha_1$ & $T^{l_2-2}\alpha_2$ & $\cdots$ & $T^{l_k-2}\alpha_k$&&&& 
\\\Cline{2-9}{.55pt}
& $T^{l_1-1}\alpha_1$ & $T^{l_2-1}\alpha_2$ & $\cdots$ & $T^{l_k-1}\alpha_k$ 
& $\alpha_{k+1}$ & $\cdots$ & $\alpha_m$ & $\ker T$ 
\\\bottomrule 
\end{tabular}
\end{center}

\end{document}

The custom rules are from:

How can I reproduce this table with thick lines?

Thick Lines on table

Answer 1

You could use the tabularray package. This makes it easy to add lines wherever you like:

\documentclass{article}

\usepackage{amsmath}
\usepackage{fourier}
\usepackage{setspace}
\setstretch{1.5}
\DeclareMathOperator{\im}{im}

\usepackage{tabularray}
\UseTblrLibrary{booktabs}

\begin{document}

\begin{center}
\begin{tblr}{
  cells={halign=c,mode=imath},
  hline{1,2,Z} = {1pt,solid},
  hline{3}={1}{1-5}{1pt,solid},
  hline{Y}={1}{2-Z}{1pt,solid},
  vline{2-X}={1pt,solid}
}
& V_1 & V_2 & \cdots & V_k & V_{k+1} & \cdots & V_m \\
& \alpha_1 & \alpha_2 & \cdots & \alpha_k &&&& \\
\SetCell[r=4]{} \im T & T\alpha_1 & T\alpha_2 & \cdots & T\alpha_k &&&& \\
& \vdots & \vdots & & \vdots &&&& V/\ker T \\
& T^{l_1-2}\alpha_1 & T^{l_2-2}\alpha_2 & \cdots & T^{l_k-2}\alpha_k&&&&  \\
& T^{l_1-2}\alpha_1 & T^{l_2-2}\alpha_2 & \cdots & T^{l_k-2}\alpha_k 
& \alpha_{k+1} & \cdots & \alpha_m & \ker T \\
\end{tblr}
\end{center}

\end{document}

Answer 2

Here is a solution with {NiceTabular} of nicematrix.

\documentclass{article}
\usepackage{multirow}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{fourier}
\usepackage{setspace}
\usepackage{nicematrix,tikz}

\NiceMatrixOptions
  {
    custom-line =
     {
       letter = " ,
       ccommand = Cline ,
       tikz = {line width = 0.65 pt} ,
       total-width = 0.65 pt
     }
  }

\setstretch{1.5}
\DeclareMathOperator{\im}{im}

\begin{document}

\begin{center}
\begin{NiceTabular}{c"c"c"c"c"c"c"c"c}\toprule 
& $V_1$ & $V_2$ & $\cdots$ & $V_k$ & $V_{k+1}$ & $\cdots$ & $V_m$ 
\\ \midrule
& $\alpha_1$ & $\alpha_2$ & $\cdots$ & $\alpha_k$ &&&& 
\\\Cline{1-5}
\multirow{4}{*}{$\im T$} & $T\alpha_1$ & $T\alpha_2$ & $\cdots$ & $T\alpha_k$ &&&& 
\\
& $\vdots$ & $\vdots$ & & $\vdots$ &&&& $V/\ker T$
\\
& $T^{l_1-2}\alpha_1$ & $T^{l_2-2}\alpha_2$ & $\cdots$ & $T^{l_k-2}\alpha_k$&&&& 
\\\Cline{2-9}
& $T^{l_1-1}\alpha_1$ & $T^{l_2-1}\alpha_2$ & $\cdots$ & $T^{l_k-1}\alpha_k$ 
& $\alpha_{k+1}$ & $\cdots$ & $\alpha_m$ & $\ker T$ 
\\\bottomrule 
\end{NiceTabular}
\end{center}

\end{document}