What's the difference between $\mathbb{R}^3$ and $\mathbb{R}_3$? My course problem booklet (mathematics BSc, second year module in algebra, unpublished) uses them in a context which suggests $\mathbb{R}_3$ might be the same as M$_{1,3}(\mathbb{R})$. I'm used to the idea that $\mathbb{R}^3$ is the home of elements which in some sense contain 3 elements from $\mathbb{R}$, but I'm not sure exactly what distinction is being made by the subscript vs. superscript.
The subscript expression occurs in part (f) of the question shown below.
Screenshot of question:

 A: I think Mathematician 42's first comment is the correct answer. I will try to elaborate on it.
First, let $$f = ax + by + cz$$
Observe that we can write
$$f = \begin{pmatrix} a & b & c \end{pmatrix} \begin{pmatrix} x \\ y \\ z\end{pmatrix} \tag{*}$$
Here we use $(\cdot)$ instead of $[\cdot]$ for row vectors but this is only a matter of convention. I think $(*)$ is the reason why the author used row vector for $a, b, c$ but column vector for $x, y, z$. This is very natural once you take matrix multiplication into consideration.
Now if we set $$\begin{pmatrix} a & b & c \end{pmatrix} = \begin{pmatrix} a_0 & b_0 & c_0 \end{pmatrix}$$ We immediately see that
$$f|_{\begin{pmatrix} a & b & c \end{pmatrix} = \begin{pmatrix} a_0 & b_0 & c_0 \end{pmatrix}}$$ is a linear function from $\mathbb{R}^3 \to \mathbb{R}$ because it maps $$\begin{pmatrix} x \\ y \\ z\end{pmatrix} \mapsto a_0 x + b_0 y + c_0 z$$
This is why the author said "the linear function $ax + by + cz$ on $\mathbb{R}^3$" in part (f).
To sum up,
$$\mathbb{R}^3 = \mathbb{R}_{3,1} = \left\{\begin{pmatrix} u \\ v \\ w\end{pmatrix} \middle| u, v, w \in \mathbb{R} \right\}$$
and
$$\mathbb{R}_3 = \mathbb{R}_{1,3} = \left\{\begin{pmatrix} u & v & w\end{pmatrix} \middle| u, v, w \in \mathbb{R} \right\}$$
A: Any answer will be somewhat speculative because we do not get the definitions of $\mathbb R^3$ and $\mathbb R_3$. Your problem booklet certainly refers to some text and I guess you will find all definitions therein. Anyway, your screenshot suggests that an obvious distinction between $\mathbb R^3$ and $\mathbb R_3$ is this: Elements of $\mathbb R^3$ are written as column vectors, elements of $\mathbb R_3$ as row vectors. Column vectors are nothing else than $(3 \times 1)$-matrices, row vectors are $(1  \times 3)$-matrices. You may regard this a mere notational issue, and in fact for the purpose of the exercise it does not play an essential role. However written, the vector $[a,b,c]$  is only used to define the linear function $\lambda(x,y,z)  =ax + by + cz$.
But recall that each linear maps $f : \mathbb R^n \to \mathbb R^m$ can be represented by a $(m \times n)$-matrix $A$. Then we get $f(x) =  A \cdot x$, where $x$ and $f(x)$ are written as column vectors and $\cdot$ denotes matrix multiplication. Using this identification of linear maps and matrices, we see that the row vector $[a,b ,c]$ can be regarded as a linear map $\lambda : \mathbb R^3 \to \mathbb R$. This gives a different interpretation of row vectors and column vectors although both are described by three real parameters.
For a more detailed discussion you can have a look at Is there a reason vectors in space are represented as column vectors (in that nothing works with row vectors)?
