What is a simple means of proving that 3 vectors belonging to $\Bbb{R}^2$ are linearly dependent? For my linear algebra class, there is a 2 part problem that asks, given the set {(1 2), (-1 -1), (1 0)}, prove or disprove that it is linearly independent using the definition only AND then prove or disprove that it is linearly independent "with a quicker proof that uses more than just the definition". For the first part I disproved that it is linearly independent by providing a counterexample, but for the second part I don't know what to do. My gut tells me that you can't have more than 2 linearly independent vectors that belong to  $\Bbb{R}^2$ (since the rref of a matrix with these vectors as columns could have at most 2 pivots), but I don't know how to express this in a proof.
If $v_1$ and $v_2$ are linearly dependent, you're done. If $v_1$ and $v_2$ are linearly independent  then $\{v_1,v_2\}$ is  a basis for $R^2$ so there are reals $x,y$ with $v_3=x v_1+y v_2.$
It turns out that what he was looking for was some quick statement based on the following proposition: Let V be an n-dimensional vector space. If {$u_1$, ... , $u_p$} $\subset$ V and p > n, then {$u_1$, ... , $u_p$} is linearly dependent.