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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?

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Yes, they must lead to the same result. The linear independence of those vectors is related to the determinant of the matrix (they are independent if and only if the determinant is different from 0) and you can prove that those matrices have the same determinant.

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