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So lets say I have the vectors: $\begin{bmatrix}0 & -1 & -2 & 1\end{bmatrix}$ and $\begin{bmatrix}3 & 1 & -1 & 0\end{bmatrix}$ and $\begin{bmatrix}2 & 1 & 5 & 1\end{bmatrix}$. Does it matter if I then use the matrix: $$ \begin{bmatrix} 0 & 3 & 2\\ -1 & 1 & 1\\ -2 & -1 & 5\\ 1 & 0 & 1\\ \end{bmatrix} $$ or the matrix: $$ \begin{bmatrix} 0 & -1 & -2 & 1\\ 3 & 1 & -1 & 0\\ 2 & 1 & 5 & 1\\ \end{bmatrix} $$ So does it matter if I use the vectors as columns or as rows?

I just used the row vectors as rows in a matrix then turned them into row echelon form and saw that they were all independent and thus said that the basis of the span of these given vectors are the vectors themself.

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If all you want to do is determine which vectors are linearly dependent, then no, it doesn't matter. You can put them in as either row vectors or column vectors, reduce, and determine the linearly independent set. This is because the row space is equal to the column space of $A^T$, and if you look at your second matrix, it is exactly the transpose of your first matrix. Therefore, its row vectors (the column vectors of your first matrix) will have the same span.

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