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Find two bases of R4 which both contain (0,1,1,1), (1,1,1,1)

by RoRi

Find two different bases for \mathbb{R}^4 which both contain the vectors (0,1,1,1) and (1,1,1,1).

Let

    \begin{align*} \mathcal{B}_1 &= \{ (1,1,1,1), (0,1,1,1), (0,0,1,1), (0,0,0,1) \} \\ \mathcal{B}_2 &= \{ (1,1,1,1), (0,1,1,1), (0,1,0,0), (0,0,1,0) \}. \end{align*}

Then \mathcal{B}_1 and \mathcal{B}_2 are both bases of \mathbb{R}^4 (it is sufficient to show they are linearly independent sets, which can be readily checked). They also both contain the requested vectors.

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