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Find two bases for R3 containing (0,1,1), (1,1,1)

Find two different bases for \mathbb{R}^3 which both contain the vectors (0,1,1) and (1,1,1).


The sets \mathcal{B}_1 = \{ (0,1,1), (1,1,1), (0,1,0) \} and \mathcal{B}_2 = \{ (0,1,1), (1,1,1), (0,0,1) \} are both bases of \mathbb{R}^3 containing (0,1,1) and (1,1,1). We can see they are bases since

    \begin{align*}  x(0,1,1) + y(1,1,1) + z(0,1,0) = (0,0,0) \quad \implies \quad x = y = z =0 \\  x(0,1,1) + y(1,1,1) + z(0,0,1) = (0,0,0) \quad \implies \quad x = y = z = 0. \end{align*}

Hence, they are each sets of three independent vectors in \mathbb{R}^3; hence, are both bases.

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