Simple Question about linear dependence

If I want to find out which of the following sets are linearly independent/dependent and I want to put it into a matrix and solve by Gaussian elimination

For the following set
4-x+6x^2, 5+2x+7x^2,1+3x+x^2

Would my matrix be [4,-1,6;5,2,7;1,3,1] [4,5,1;-1,2,3;6,7,1] (also each row would have a 0 at the end of the row)

I originally thought it was the first matrix as thats what books and youtube videos have told me such as this one. However my teacher said this was wrong and I should have used the second matrix
http://www.youtube.com/watch?v=8mavvyGOedM

In the first case, you're trying to find the maximum number of linearly independent rows of the matrix:

$A = \begin{pmatrix} 4 & -1 & 6 \\ 5 & 2 & 7 \\1 & 3 & 1 \end{pmatrix}$

In the second case, you're trying to find the maximum number of linearly independent rows of the matrix:

$B = \begin{pmatrix} 4 & 5 & 1 \\ -1 & 2 & 3 \\6 & 7 & 1 \end{pmatrix}$

However, $A = B^{T}$ so the rows of $A$ are the columns of $B$, and vice versa.

It is a theorem in linear algebra that, for any matrix, the number of linear independent rows (the row rank) is equal to the number of linearly independent columns (the column rank). So if it turns out that the rows of $A$ are linearly independent, then it will be true that the columns of $A$ are also linearly independent, but since $A = B^{T}$, that means that the rows of $B$ are linearly independent (and vice versa, of course).

So it won't make any difference which matrix you use to test for linear independence.

Im sorry I've write this out wrong, It should have been: Would my matrix be [4,-1,6;5,2,7;1,3,1] OR [4,5,1;-1,2,3;6,7,1] (also each row would have a 0 at the end of the row). And then I use row reduction to try and get it in the echelon form, and see if it has non trivial solutions.
In this case would it make a difference which matrix it would be?

Maybe I'm being dimwitted, but haven't you just repeated the same question?