Find linear transformation given kernel

Let $u=(1,2,3,4)$ and $v=(0,1,1,1)$. Clearly, $(u,v)$ is an independent family of $\mathbb{R}^4$, and can be completed to a basis of $\mathbb{R}^4$ with two vectors. It's very easy to see that adding $e_1=(1,0,0,0)$ and $e_2=(0,1,0,0)$ to this family yields a basis of $\mathbb{R}^4$ (compute the rank of $(u,v,e_1,e_2)$).

Now a linear mapping is uniquely determined by the image of a basis of its domain. We already know the image of $u$ and $v$ (the nil vector of $\mathbb{R}^3$), hence we only need to determine the image of $e_1$ and $e_2$ by $F$. A bit later on, we'll need the coordinates of a vector of $\mathbb{R}^4$ in the basis $(u,v,e_1,e_2)$, so we might as well do this right now. A straightforward system solving yields: $$\forall (x,y,z,t)\in\mathbb{R}^4,\quad (x,y,z,t)=(-z+t)u+(4z-3t)v+(x+z-t)e_1+(y-2z+t)e_2.$$

There's a caveat: we can't take just any vectors $a$ and $b$ of $\mathbb{R}^3$ for the images of $e_1$ and $e_2$, since we could create some other vectors in the kernel of $F$. The restriction is that the family $(a,b)$ should be independent (think of the Rank–Nullity Theorem).

We then have the general solution of your problem: a linear mapping $F:\mathbb{R}^4\longrightarrow\mathbb{R}^3$ satisfies all your requirements if and only if there exists an independent family $(a,b)$ of vectors of $\mathbb{R}^3$ such that: $$\forall(x,y,z,t)\in\mathbb{R}^4,\quad F(x,y,z,t)=(x+z-t)a+(y-2z+t)b.$$

For example, the linear mapping $F:\mathbb{R}^4\longrightarrow\mathbb{R}^3$ defined by: $$\forall(x,y,z,t)\in\mathbb{R}^4,\quad F(x,y,z,t)=(x+z-t,y-2z+t,0)$$ fulfills all your requirements. The matrix of this $F$ in the standard bases of $\mathbb{R}^4$ and $\mathbb{R}^3$ is: $$[F]_{\text{std}(\mathbb{R}^4),\text{std}(\mathbb{R^3})}=\begin{pmatrix}1&0&1&-1\\0&1&-2&1\\0&0&0&0\end{pmatrix}.$$

But there are lots more (I actually gave you all of them!).

The example given by YourAdHere corresponds to the case $a=(3,3,0)$ and $b=(1,2,0)$ (which is, in some respect, not the simplest).

Now, given a $3\times4$ matrix, how can you determine whether its the matrix (in the standard bases of $\mathbb{R}^4$ and $\mathbb{R}^3$) of a linear mapping satisfying your requirements? Easy: