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Tabular custom rules across certain columns

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

enter image description here

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

enter image description here

\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*

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