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Then we represent all pairs $(P, Q)$ of projections such that $T=\\lambda P+Q$ for a fixed $\\lambda$, and find that all such pairs are connected if $\\lambda\\in\\mathbb{R}\\backslash\\{-1, 0, 1\\}$. Afterwards, the von Neumann algebra generated by such pairs $(P,Q)$ is characterized. 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We give a sufficient and necessary condition for $T$ to be the pencil $\\lambda P+Q$ of a pair $( P, Q)$ of projections at some point $\\lambda\\in\\mathbb{R}\\backslash\\{-1, 0\\}$. Then we represent all pairs $(P, Q)$ of projections such that $T=\\lambda P+Q$ for a fixed $\\lambda$, and find that all such pairs are connected if $\\lambda\\in\\mathbb{R}\\backslash\\{-1, 0, 1\\}$. Afterwards, the von Neumann algebra generated by such pairs $(P,Q)$ is characterized. 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