I'm very confused by the quotient linear transformation, and I'll try and illustrate my confusions in the following example:
I have no problem showing (a), however I have serious conceptual difficulties understanding (b). Does anyone know how I could go about thinking about what the question is asking? Or, what I must do? I hardly even know where to start. I know I ought to try and not just ask for an answer and give a reasoned attempt at solving this, but I just don't know where to start. My attempt was simply this:
And after "require $v$ s.t..." I don't know where to go, and I'm sure I didn't do that right.
As the dimension of quotient is 2 it suffices to show that the two given vectors are linearly independent.
If V is a vector space and $W$ is a vector subspace here is (the canonical) way to show the elements $[v_1], [v_2]\ldots $ to be linearly independent in the quotient $V/W$:
One has to show the for elements $v_1,v_2,\ldots$ in $V$ no linear combination lies in $W$. That is their span intersects $W$ only at $\{0\}$.
In this case (using more convenient row notation) you have to show that $(0,1,0,0)$ and $(0,0,1,0)$ span does contain any vector in the kernel of $T$. This is straightforward.