Show that the vectors $\begin{bmatrix} 1 \\ 2 \\ 0 \\ 2 \end{bmatrix}\begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}\begin{bmatrix} 2 \\ 0 \\ 1 \\ 3 \end{bmatrix}$ are linearly indepedent. Find a fourth vector so that the set is linearly independent.
Solution:
I row reduced a matrix composed of these vectors and found the columns were linearly independent.
To find a 4th vector so that the set is linearly independent - I picked a random vector, say, $ V= \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$
I then applied the Gram Schmidt process - I subtracted from V the projection of V onto each of the original vectors. Is this the correct way to find a linearly independent 4th vector?
My final answer was $\begin{bmatrix} \frac{17}{63} \\ \frac{-35}{63} \\ \frac{-30}{63} \\ \frac{-41}{63} \end{bmatrix}$