TIFR-2012-Maths-A: 6

True/False Question: Let $u_{1},u_{2},u_{3},u_{4}$ be vectors in $\mathbb{R}^{2}$ and $u=\sum_{j=1}^{4}t_{i}u_{j}\:\:\: ; \:\:\:t_{j}> 0 \:\: and \sum_{j=1}^{4}t_{j}=1.$ Then three vectors $v_{1},v_{2},v_{3}\in \mathbb{R}^{2}$may be chosen from $\left \{ u_{1},u_{2},u_{3},u_{4} \right \}$ such that $u=\sum_{j=1}^{3}s_{i}v_{j}, \:\:\:s_{j}\geq 0 ,\:\: \sum_{j=1}^{3}s_{j}=1.$
True/False Question: Let $u_{1},u_{2},u_{3},u_{4}$ be vectors in $\mathbb{R}^{2}$ and $u=\sum_{j=1}^{4}t_{i}u_{j}\:\:\: ; \:\:\:t_{j}> 0 \:\: and \sum_{j=1}^{4}t_{j}=1.$ Then three vectors $v_{1},v_{2},v_{3}\in \mathbb{R}^{2}$may be chosen from $\left \{ u_{1},u_{2},u_{3},u_{4} \right \}$ such that $u=\sum_{j=1}^{3}s_{i}v_{j}, \:\:\:s_{j}\geq 0 ,\:\: \sum_{j=1}^{3}s_{j}=1.$

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Aug 30, 2020
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soujanyareddy13
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