1. **Problem Statement:** Find the standard matrix $A$ representing the linear operator $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $$L\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}x + 2y \\ x + y\end{pmatrix}.$$ 2. **Formula and Explanation:** The standard matrix $A$ of a linear operator $L$ is given by $$A = \big(L(\mathbf{e}_1), L(\mathbf{e}_2)\big),$$ where $\mathbf{e}_1 = \begin{pmatrix}1 \\ 0\end{pmatrix}$ and $\mathbf{e}_2 = \begin{pmatrix}0 \\ 1\end{pmatrix}$ are the standard basis vectors in $\mathbb{R}^2$. 3. **Calculate $L(\mathbf{e}_1)$:** $$L\begin{pmatrix}1 \\ 0\end{pmatrix} = \begin{pmatrix}1 + 2\cdot0 \\ 1 + 0\end{pmatrix} = \begin{pmatrix}1 \\ 1\end{pmatrix}.$$ 4. **Calculate $L(\mathbf{e}_2)$:** $$L\begin{pmatrix}0 \\ 1\end{pmatrix} = \begin{pmatrix}0 + 2\cdot1 \\ 0 + 1\end{pmatrix} = \begin{pmatrix}2 \\ 1\end{pmatrix}.$$ 5. **Form the matrix $A$:** $$A = \begin{pmatrix}1 & 2 \\ 1 & 1\end{pmatrix}.$$ **Final answer:** The standard matrix representing $L$ is $$\boxed{\begin{pmatrix}1 & 2 \\ 1 & 1\end{pmatrix}}.$$