For the subspace $$W = \{(x_1 , x_2 , x_3, x_4 ) ∈ \Bbb R^4 : x_1 + x_2 = 0, x_3 + x_4 = 0\}$$ of vector space $V=\Bbb R^4$ over $\Bbb R$. Find the basis of quotient space $V/W$.
Quotient space can be stated as follows: Let $V$ be a vector space over a field $F$, and let $W$ be a subspace of $V$. The set $V /W$ is the set defined by
$V /W = \{v + W : v\in V\}$
That is, $V /W$ is the collection of cosets of $W$ in $V$.
My approach: One basis of the solution space of subspace $W$ consisting of vectors $(-1, 1, 0, 0)$ and $( 0, 0, -1, 1)$. This can be extended to basis of $V$ by adding additional vectors $v_1 = (0, 0, 1, 0)$ and $v_2 = (0, 1, 0, 0)$. Then the basis for $V /W$ is $v_1+ W$ and $v_2 +W$.
Am I correct. Is there any error in my solution or approach?
I edited
Thanks
Looks fine to me.
(The following was written in response to an earlier draft of the Question.)
The span of $(-1,1,0,0)$ includes $(1,-1,0,0)$, so these two vectors are linearly dependent -- no basis contains both of them. You have a good idea, but you have ignored the constraint on $x_3$ and $x_4$. If you correct this omission, I think you have the right idea and follow-through.