If $U$ is a subspace of $\mathbb{R}^3$ spanned by the vectors $\{(2,0,-1),(1,2,0),(0,4,1)\}$, how can I find the basis for $\mathbb{R}^3 / U$?
I'm having trouble understanding how one can define a basis for all the vectors in $\mathbb{R}^3$ which doesn't span specifically the vectors in $U$. I think it'd be possible to find some cartesian coordinates for $U$ and then replace the $=$ sign for $\ne$, but that doesn't seem to help much. So far, I've found that the three vectors that span $U$ aren't linearly independent, but they can be used to find the basis $\{(1,2,0),(0,4,1)\}$. Therefore, one can get the parametric equations of $U$, and then its cartesian equation: $2x-y+4z=0$.
Can someone hint me in the right direction?
Hint You can find a vector $v_3$ such that $\{(1,2,0),(0,4,1), v_3\}$ is a Basis in $\mathbb R^3$. Then you can easily argue that $\{ v_3+U \}$ is a basis for $\mathbb R^3/U$.
This can be done in various ways: wither finding a basis among the vectors $\{(1,2,0),(0,4,1), (1,0,0), (0,1,0), (0,0,1)\}$ which contains the first two vectors, by G-S, or by simply taking $$v_3=(1,2,0) \times (0,4,1)$$