## 4 Simplify Problem by Using Linear Equivalence  
Let \(\mathbf{M}\in\mathbf{R}^{m\times m}\) be the square matrix established by taking m-column vectors in \(\mathbf{W}\) and \(\mathbf{H}\in\mathbf{R}^{m\times(n-m)}\) consist of the remaining column vectors in \(\mathbf{W}\). According to Theorem 1, if \(\mathbf{x}\in\mathbf{WCW}\subset\mathbf{R}^{m}\) then \(\mathbf{\omega}\) must contain m linearly independent vectors, i.e., there exist a certain set of m-column vectors in \(\mathbf{W}\) that form the nonsingular matrix \(\mathbf{M}\). **Theorem 2**. : _Given that \(\mathbf{N}\in\mathbf{R}^{m\times m}\) is the non-singular matrix, if \(\mathbf{\omega}\) non-negatively spans \(\mathbf{R}^{m}\) then \(\mathbf{N}.\mathbf{\omega}=\{\mathbf{N}\mathbf{w}_{1},\mathbf{N}\mathbf{w}_{2}, \,\ldots,\,\mathbf{N}\mathbf{w}_{n}\}\) also non-negatively spans \(\mathbf{R}^{m}\). If \(\mathbf{\omega}\) does not nonnegatively span \(\mathbf{R}^{m}\), then \(\mathbf{N}.\mathbf{\omega}\) also does not non-negatively span \(\mathbf{R}^{m}\)._  
Proof of Theorem 2.: The proof of the first part of Theorem 2 can be derived from the proof of Theorem 4.6 by [PERSON] in [21]. The minimum condition in order for \(\mathbf{\omega}\) nonnegatively span \(\mathbf{R}^{m}\) is that \(\mathbf{\omega}\) must contain at least \(m+1\) affinely independent vectors. Hence, let consider \(\{\mathbf{w}_{1},\,\mathbf{w}_{2},\,\ldots,\,\mathbf{w}_{m+1}\}\) be the set of \(m+1\) affinely independent vectors in \(\mathbf{\omega}\). Suppose \(\{\mathbf{N}\mathbf{w}_{1},\,\mathbf{N}\mathbf{w}_{2},\,\ldots,\,\mathbf{N} \mathbf{w}_{m+1}\}\) is affinely dependent then \(\exists i\in\{1,\ldots,\,m+1\}\) such that \(\mathbf{N}\mathbf{w}_{i}=\sum_{j=\{1,\ldots,i-1,i\},\ldots,\,m\}\mathbf{N} \mathbf{w}_{j}\). Since \(\mathbf{N}\) is invertible, we have \(\mathbf{N}^{-1}\mathbf{N}\mathbf{w}_{i}=\mathbf{N}^{-1}\sum_{j=\{1,\ldots,i-1, i\},\ldots,m\}\mathbf{N}\mathbf{w}_{j}\). Then, by simplifying both sides, we have \(\mathbf{w}_{i}=\sum_{j=\{1,\ldots,i-1,i+1,\ldots,m\}}\mathbf{w}_{j}\). However, it violates the assumption in Theorem 2. Following the procedure analogous to that employed for the proof of the second part. \(\square\)  
According to Theorem 2, we can state that if \(\mathbf{\omega}\) does or does not non-negatively span \(\mathbf{R}^{m}\) than \(\mathbf{M}^{-1}\mathbf{\omega}\) also does or does not nonnegatively span \(\mathbf{R}^{m}\). Thus, Equation (3) can be replaced by any of the Equations (12)-(14), but the condition represented by (9) is not influent. \[\mathbf{M}^{-1}\big{[}\ \ \mathbf{M}\ \ \mathbf{H}\ \ \big{]}\mathbf{t}=\mathbf{M}^{ -1}\psi_{e} \tag{12}\]  
\[\big{[}\ \ \mathbf{I}\ \ \ \mathbf{M}^{-1}\mathbf{H}\ \ \big{]}\mathbf{t}= \mathbf{M}^{-1}\psi_{e} \tag{13}\]  
\[\mathbf{It}_{A}+\mathbf{M}^{-1}\mathbf{H}\mathbf{t}_{B}=\mathbf{M}^{-1}\psi_{e} \tag{14}\]  
where \(\mathbf{t}_{A}=[\mathbf{t}_{1}\ \mathbf{t}_{2}\ \ldots\ \mathbf{t}_{m}]^{T}\) and \(\mathbf{t}_{B}=[\mathbf{t}_{m+1}\ \mathbf{t}_{m+2}\ \ldots\ \mathbf{t}_{n}]^{T}\). Based on Equation (12), we can write \(\mathbf{M}^{-1}\mathbf{\omega}=[\mathbf{e}_{1},\mathbf{e}_{2},\,\ldots,\,\mathbf{e}_{m},\, \mathbf{M}^{-1}\mathbf{w}_{m+1},\,\ldots,\,\mathbf{M}^{-1}\mathbf{w}_{n}]\), where \(\{\mathbf{e}_{1},\,\mathbf{e}_{2},\,\ldots,\,\mathbf{e}_{m}\}\) is the standard basis of \(\mathbf{R}^{m}\). Following the Equation (10), the set of m-first column vectors in \(\mathbf{W}\) is now represented as \(\mathbf{\omega}_{A}=[\mathbf{e}_{1},\,\mathbf{e}_{2},\,\ldots,\,\mathbf{e}_{m}]\) and the set of remaining vectors is \(\mathbf{\omega}_{B}=[\mathbf{M}^{-1}\mathbf{w}_{m+1},\,\ldots,\,\mathbf{M}^{-1} \mathbf{w}_{n}]\). Geometrically, \(\mathbf{\omega}_{A}\) non-negatively spans closed nonnegative hyperoctant of \(\mathbf{R}^{m}\), \(\mathbf{R}^{m}_{+}\). Hence, if \(\mathbf{x}\) belongs to WCW, then the intersection between the interior of cone \(\mathbf{\omega}_{B}\) and open negative hyperoctant \(\mathbf{R}^{m}_{-}\) should not be empty.