Question
Let W be the subspace of $R^5$ spanned by
$ u_1
= (1, 2, –1, 3, 4)\\
u_2 = (2, 4, –2, 6, 8) \\
u_3 = (1, 3, 2, 2, 6)\\ u_4 = (1, 4, 5, 1, 8)\\u_5 = (2, 7, 3, 3, 9)$
Find a subset of the vectors which form a basis of W.
Doubt
Now. I understand that, I need to make a matrix and by doing row operations have to convert it to reduced echelon form, and the zero rows suggest the Linearly dependent(LD) vectors, but I don't think this is a correct method because, after few operations, when I see 2 rows of matrix which looks scalar multiple of each other, it's onto me that which row I want to make zero(true, isn't it ?),so it will change the answer
Your reasoning is correct, it doesn't matter which rows you choose to subtract and zero out, as long as you follow the operations of Gaussian elimination and arrive at the row-echelon form. Bases don't need to be unique. You're good as long as they span the same space.
Put all your vectors in a matrix: $$ \begin{bmatrix} {1} & 2 & 1 & 1 & 4\\ {2} & 4 & 3 & 4 & 8 \\ {-1} & -2 & 2 & 5 & 6 \\ {3} & 6 & 2 & 1 & 8 \\ {4} & 8 & 6 & 8 & 9\\ \end{bmatrix} $$ this should reduce to:
\begin{bmatrix} {1} & 2 & 0 & -1 & 0\\ {0} & 0 & 1 & 2 & 0 \\ {0} & 0 & 0 & 0 & 1 \\ {0} & 0 & 0 & 0 & 0 \\ {0} & 0 & 0 & 0 & 0\\ \end{bmatrix}
Pick the vectors that correspond to the pivot columns to get the following basis: $B = \{ u_1, u_3, u_5\}$