Normally when testing if two or more vectors are linearly independent, we would first put them in row echelon form and perform Gauss elimination on them.
Here are the vectors $[a_1, a_2, a_3]$, $[b_1, b_2, b_3]$ and $[c_1, c_2, c_3]$ in row echelon form, ready for Gauss elimination:
\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{bmatrix}
Is it possible to determine if they are linearly independent by instead Gauss-eliminating the following matrix:
\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{bmatrix}
Are the processes equivalent?