Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly dependent.
If the vectors in $S$ are linearly dependent, each vector in $S$ can be written as linear combinations of the other vectors. If add a vector $v_4$, and multiply this vector by a scalar $k=0$, then the vectors $\{v_1,v_2,v_3\}$ can still be written as linear combinations of each other.
Is this a proof?