If ${\bf X}$ is an $n\times p$ real matrix, the column space or range of ${\bf X}$ is defined to be the set spanned by its columns:
$$
Col\left({\bf X}\right)=\left\{ {\bf x}\in\mathbb{R}^{n}\,:\,\exists\boldsymbol{\beta}\in\mathbb{R}^{p},{\bf x}={\bf X}\boldsymbol{\beta}\right\} .
$$
Assume the vector ${\bf x}_{i}^{T}=\left[x_{1i},...,x_{ik}\right]$, and constraint ${\bf x}_{1}+{\bf x}_{2}+{\bf x}_{3}={\bf 1}$. Define matrices
\begin{align*}
{\bf X}^{T} & =\left[\begin{array}{c}
{\bf x}_{1}^{T}\\
{\bf x}_{2}^{T}
\end{array}\right]\\
\tilde{{\bf X}}^{T} & =\left[\begin{array}{c}
{\bf x}_{2}^{T}\\
{\bf 1}-{\bf x}_{1}^{T}-{\bf x}_{2}^{T}
\end{array}\right].
\end{align*}
What are the conditions such that $Col\left({\bf X}\right)=Col\left(\tilde{{\bf X}}\right)$ ?
ผมพยายามจะหา conditions ที่ $Col\left({\bf X}\right)=Col\left(\tilde{{\bf X}}\right)$ มันเท่ากันอะครับ มันดูเหมือนไม่มีอะไร แต่ผมคิดไม่ออก รบกวนพี่ๆทุกท่านช่วยแนะนำหน่อยครัย
