Suppose $v1, v2, v3$ are vectors in $R^n$, and that span($v1,v2,v3$) = span($v1,v2$).
How come ($v1,v2,v3$) cannot be linearly independent?
I understand that the definition of a linearly independent set of vectors is that $a_1v_1+ ... +a_kv_k = 0$ only when $a_1= ...= a_k = 0$, but I am failing to see what part of this theorem is specifically limiting those spans from being equal.
Because $v_3\in$ span$(v_1,v_2,v_3)$, so $v_3\in$ span$(v_1,v_2)$, which means there exists a linear combination of $v_1,v_2$ that is $v_3$, contradicting linear independence.