Linear Mathematics | Part IB, 2003

Let a1,a2,,ana_{1}, a_{2}, \ldots, a_{n} be distinct real numbers. For each ii let vi\mathbf{v}_{i} be the vector (1,ai,ai2,,ain1)\left(1, a_{i}, a_{i}^{2}, \ldots, a_{i}^{n-1}\right). Let AA be the n×nn \times n matrix with rows v1,v2,,vn\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} and let c\mathbf{c} be a column vector of size nn. Prove that Ac=0A \mathbf{c}=\mathbf{0} if and only if c=0\mathbf{c}=\mathbf{0}. Deduce that the vectors v1,v2,,vnspanRn\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} \operatorname{span} \mathbb{R}^{n}.

[You may use general facts about matrices if you state them clearly.]

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