A real linear space $W$ of dimension $2$ is considered, with a basis $B = \{w_1, w_2\}$.
(a) Find the expression $f(x)$ of the linear mapping $f : \mathbb{R^4} \rightarrow W$ whose kernel is the subspace $N = \{(x_1, x_2, x_3, x_4) : x_1 + 3x_2 + x_4 = 0\}$ and satisfies the condition $f(0, 0, 0, 1) = \vec{w_1} − \vec{w_2}$.
(b) Discuss the equation $f(\vec{x})=\vec{y}$
I have already done part (a), just need (b). I got for the expression of $f(x)$ to be: $$f(-3,1,0,0)=(0,0,0,0)$$ $$f(-1,0,0,1)=(0,0,0,0)$$ $$f(0,0,1,0)=(0,0,0,0)$$ $$f(0,0,0,1)=\vec{w_1}-\vec{w_2}$$
And for part b) they ask us to discuss $$\begin{pmatrix} -3 & 1 & 0 & 0 & | & y_1\\ -1 & 0 & 0 & 1 & | & y_2 \\ 0 & 0 & 1 & 0 & | & y_3 \\ 0 & 0 & 0 & 1 & | & y_4 \end{pmatrix}$$
Which I'm guessing you can't do reduction with Gauss that easily but instead you can have the case where $\vec{y}=0$, which implies that the equation has a solution and it is the kernel, but for the rest I have no clue?
Let's clear some things up first. $f$ is a function $\mathbb{R}^4 \to W$. $W$ is not a set of pairs of numbers, but since it's a real vector space of dimension 2, there is a natural mapping $W\to\mathbb{R}^2$. (This is a common thing we do when considering finite dimensional vector spaces, and it's usually implicit, which can make things more confusing.)
Then we can consider $f$ by its corresponding matrix - this matrix has dimension $2\times 4$ - (why?)
Now suppose we have the opposite task - how do we find the kernel of a given matrix? The most common way is putting it in RREF. Then we read the rows of the RREF matrix to give the equations defining the kernel. In our case, this equation is $$x_1+3x_2+x_4 = 0,$$ so our RREF matrix would look like $$\begin{pmatrix} 1 & 3 & 0 & 1 \\ 0 & 0 & 0 & 0\end{pmatrix}.$$
This would work as a potential function were it not for the final condition $f(0,0,0,1) = w_1-w_2$. In terms of our matrix, this means the final column should be $$\begin{pmatrix} 1 \\ -1\end{pmatrix}$$ (again, check you know why this is the case - you may want to remind yourself of how matrix elements are defined).
We can use row operations to force this to hold. Hopefully you can work out which ones, to get the unique form of the matrix. Then it's a short hop to finding $f(x_1,x_2,x_3,x_4)$.