I am trying to solve Problem 15 on page 107 of Brešar's Uvod v Algebro. The question translates into English as:
Explain why any endomorphism of the algebra $M_2(\mathbb{R})$ can't map the matrix $E_{11}$ to the matrix $E_{12}$.
In this question, what does the notation $E_{ij}$ mean?
The notation is explained at the top of page 35 of the same text. In this text, $E_{ij} \in M_{2\times 2}(\mathbb{R})$ is the standard basis matrix with a $1$ in the $j$-th column of the $i$-th row, and $0$s everywhere else. There are four such matrices: $$ E_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\quad E_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\quad E_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},\quad\text{and}\quad E_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}. $$ In (linear) algebra and analysis, it is common to use the notation $e_i$ (or $e^{(i)}$) to denote the $i$-th standard basis vector in $\mathbb{R}^n$. That is, $e_i$ is a vector of length $n$ with a $1$ in the $i$-th position and $0$s everywhere else, i.e. $$ e_1 = (1, 0, \dotsc, 0),\qquad e_2 = (0, 1, \dotsc, 0),\qquad\dotsc,\qquad e_n = (0, 0, \dotsc, 1). $$ Every vector in $\mathbb{R}^n$ can be written as a linear combination of these vectors, so they give us a natural way of decomposing general vectors into simpler objects. Similarly, every matrix in $M_{2\times 2}(\mathbb{R})$ may be written as a linear combination of the $E_{ij}$s, hence the author's adoption of this notation—using a capital $E$ to distinguish matrices from vectors—is reasonable and not entirely unheard of.
It may also be worth noting, as per copper.hat's comment, that $$ E_{ij} = e_{i}^\intercal e_{j}, $$ where $e_{i}^\intercal$, a column vector, is thought of as an $m\times 1$ matrix; and $e_{j}$, a row vector, is thought of as a $1\times n$ matrix.