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

by RoRi

April 28, 2016

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

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