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Given that $$U = \{\alpha i + \beta k | α,β ∈\mathbb R\}$$ Write down two linearly independent vectors $u, v$ in this set.

Show that if you take any third vector $w ∈ U$ ,then the three vectors $u, v, w$ are linearly dependent.

My half attempt: $$\mathbf v = \{0\mathbf i + 1\mathbf k\} $$ $$\mathbf u = \{1\mathbf i + 0\mathbf k\} $$

I'm basically stuck here. Any help/clarification is greatly appreciated (: Side note: how do you go about proving this without using matrices?

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  • $\begingroup$ How are $i$ and $k$ defined? $\endgroup$ – Michael Hoppe Mar 23 '17 at 10:23
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So it seems like $U$ has dimension 2. Meaning that two linearly independent vectors form a basis. Hence, any third vector can be written as a linear combination of your two basis elements. Maybe you can work your way from here.

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