# Proving that given two linearly independent vectors in a plane, any third vector in that same plane is linearly dependent

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?

• How are $i$ and $k$ defined? – Michael Hoppe Mar 23 '17 at 10:23

## 1 Answer

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.