Linear dependence of 3 vectors in $\mathbb{R}^4$ Let $a,b,c \in \mathbb{R},$ $\vec{v_1}=\begin{pmatrix}1\\4\\1\\-2 \end{pmatrix},$ $\vec{v_2}=\begin{pmatrix}-1\\a\\b\\2 \end{pmatrix},$ and $\vec{v_1}=\begin{pmatrix}1\\1\\1\\c \end{pmatrix}.$ What are the conditions on the numbers $a,b,c$ so that the three vectors are linearly dependent on $\mathbb{R}^4$? I know that the usual method of solving this is to show that there exists scalars $x_1,x_2,x_3$ not all zero such that
\begin{align}
x_1\vec{v_1}+x_2\vec{v_2}+x_3\vec{v_3}=\vec{0}.
\end{align}
Doing this would naturally lead us to the augmented matrix
\begin{pmatrix}
  1 & -1 & 1 &0\\
  4 & a & 1 &0\\
  1& b & 1 &0\\
  -2 & 2 & c &0\\
\end{pmatrix}
Doing some row reduction would lead us to the matrix
\begin{pmatrix}
  1 & -1 & 1 &0\\
  4 & a & 1 &0\\
  0& b+1 & 0 &0\\
  0 & 0 & c+2 &0\\
\end{pmatrix}
I'm not quite sure how to proceed after this. Do I take cases on when whether $b+1$ or $c+2$ are zero and nonzero? 

The matrix
$$
\begin{pmatrix}
  1 & -1 & 1 &0\\
  4 & a & 1 &0\\
  0& b+1 & 0 &0\\
  0 & 0 & c+2 &0\\
\end{pmatrix}
$$
is not in row echelon form. Sum to the second column the first multiplied by $-4$, getting
$$
\begin{pmatrix}
  1 & -1 & 1 &0\\
  0 & a+4 & -3 &0\\
  0& b+1 & 0 &0\\
  0 & 0 & c+2 &0\\
\end{pmatrix}
$$
Still not row echelon form, but “almost”. Anyway, computing the rank is easy.
If $b+1\ne0$, you can swap the second and third row, then continue the Gaussian elimination getting
$$
\begin{pmatrix}
  1 & -1 & 1 & 0\\
  0 & 1 & 0 & 0\\
  0& 0 & 1 & 0\\
  0 & 0 & 0 & 0\\
\end{pmatrix}
$$
which has rank $3$. Thus a necessary condition so the vectors are linearly dependent is $b=-1$. Now the matrix becomes
$$
\begin{pmatrix}
  1 & -1 & 1 &0\\
  0 & a+4 & -3 &0\\
  0& 0 & 0 &0\\
  0 & 0 & c+2 &0\\
\end{pmatrix}
$$
If $a+4\ne0$ and $c+2\ne0$, a row echelon form is (swapping the third and fourth rows)
$$
\begin{pmatrix}
  1 & -1 & 1 &0\\
  0 & a+4 & -3 &0\\
  0 & 0 & c+2 &0\\
  0& 0 & 0 &0\\
\end{pmatrix}
$$
so the matrix has rank $3$. In order the rank is less than $3$ we need either $a=-4$ or $c=-2$.
You can so state that the vectors are linearly dependent if and only if $b=-1$ and either $a=-4$ or $c=-2$.
