I have a vector space $L$, which is a subspace of $\mathbb{R}^4$, spanned by these vectors: $(4, 1, 1, 2), (2, 3, 1, 0), (-10, 35, 5, -20), (2, 13, 3, -4).$
I need to find two vectors from $\mathbb{R}^4$ which don't belong to $L$.
I attempted to solve this by firstly reducing the $L$ matrix.
I found that $L$ has a dimension of $2$.
Then, I took the $2$ linearly independent vectors from the reduced matrix: $(1, 0, -10, -2), (0, 1, 15, 5),$ and tried finding $3^\text{rd}$ and $4^\text{th}$ vectors with which the $2$ vectors would still be linearly independent.
I found that $2$ such vectors are $(0, 0, 1, 0) \text{ and } (0, 0, 0, 1).$
Did I get this right?