Creating a single layer perceptron for the OR problem

I am working on the following problem

Find the linear least squares unit weights for the `OR' problem, ie. $v_1^T = (0,0), v_2^T = (1,0), v_3^T = (0,1), v_4^T = (1,1)$ and $u_1 = 0, u_2 = u_3 = u_4 = 1$.

Here $v$ represent inputs and $u$ outputs. For problems like this I usually find a matrix $W$ (the weights) such that $$u_i=Wv_i \quad i=1,2,3,4$$ but I think it is obvious a matrix doesn't exist. my reasoning being that the problem is equivalent to finding the matrix $W = \begin{pmatrix} x_1 & x_2 \end{pmatrix}$ such that $$\begin{pmatrix} 0 & 1 & 1 & 1 \end{pmatrix} = \begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix}$$ And looking at the two middle columns of the left vector and right matrix we must have $x_1=x_2 =1$ but then we get $1 + 1 = 1$, a contradiction.

In a unit perceptron a function may be applied to the output, and a step function $$f(x) \begin{cases} 0 & x \leq 0 \\ 1 & x > 0 \end{cases}$$ would work here.

My question here is, is this correct? I wasn't quite sure if I am answering the right question. I am pretty sure $u_i = f(Wv_i)$ but I am not too familiar with what is meant by "linear least squares unit weights" in this context?

The fact that you mention linear least squares error seems to hint that they want you to use a completely linear model $u_i=Wv_i$ for your perceptron. In this case you won't get exact answers like $0$ and $1$; you will only get approximations with some amount of error.
That said, I don't think a linear model makes sense for this problem, since it is a classification problem with two classes. Using a non-linear step-like function $f$ before the final output of a perceptron classifier is a pretty standard thing to do, so I see no problem with using $u_i=f(Wv_i)$.
Assuming you can use $f$, using the weights $x_1 = x_2 = 1$ as you mentioned solve this problem exactly with zero error.