making equality notation below each other in multi lines equations

Question

how to make the first and one before last equality sign below each other in:

\begin{multline}
\langle z,T\rangle=  z_{0}T_{0}+...+z_{n-1}T_{n-1}+1 \\
=z_{0}T_{0}+...+z_{n-1}T_{n-1}+k(T_{0}+...+T_{n-1})+1 = (z_{0}+k)T_{0}+...+(z_{n-1}+k)T_{n-1}+1\\ =z_{0}^{\prime}T_{0}+...+z^{\prime}_{n-1}T_{n-1}+1=  \langle z^{\prime}, T\rangle
\end{multline}

Answer 1

For a single alignment point across multiple equations, use align:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align}
  \langle z, T \rangle &= z_0 T_0 + \dots + z_{n-1} T_{n-1} + 1 \nonumber \\
                       &= z_0 T_0 + \dots + z_{n-1} T_{n-1} + k (T_0 + \dots + T_{n-1}) + 1 \nonumber \\
                       &= (z_0 + k) T_0 + \dots + (z_{n-1} + k) T_{n-1} + 1 \nonumber \\
                       &= z_0^\prime T_0 + \dots + z^\prime_{n-1} T_{n-1} + 1 = \langle z^\prime, T \rangle
\end{align}

\end{document}