Calculating the reduced row echelon form to find the maximum set of linearly independent vectors.

Question

I calculated the reduced row of the following matrix:

$$\begin{equation*} \mathbf{X} = \left( \begin{array}{ccc} 2 & 0 & 4 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 2 & 1 \\ 0 & 2 & 4 & 1 \end{array} \right) \end{equation*}$$

And then I got the following matrix $$\begin{equation*} \mathbf{X} = \left( \begin{array}{ccc} 1 & 0 & 2 & 1/2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array} \right) \end{equation*}$$

But then I want to know what are my independent vectors (the maximum number of them ), so what shall I do? (I am given a set of vectors that are not linearly independent and I am trying to find the maximum set of linearly independent vectors of them to use them in Gram-Schmidt)

Answer 1

But then I want to know what are my independent vectors (the maximum number of them ), so what shall I do?

Well, since echelon form contains all-zeros in the last row, then the rank is $4 - 1 = 3$, then to extract the linearly independent vectors, just take the first three rows. Actually, any combination of $3$ rows would do ;)