Suppose L: \mathbb{R}^2 \to \mathbb{R}^2 is a linear transformation. Let \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} be the standard basis vectors in \mathbb{R}^2 and suppose that L(\mathbf{e}_1) = \begin{bmatrix} 1 \\ -1 \end{bmatrix} and L(\mathbf{e}_2) = \begin{bmatrix} 3 \\ 5 \end{bmatrix}. Let \mathbf{x} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}. Find L(\mathbf{x}). Hint: Write \mathbf{x} as a linear combination of the standard basis vectors: \mathbf{x} = \begin{bmatrix} 2 \\ 4\end{bmatrix} = 2 \mathbf{e}_1+4\mathbf{e}_2. Since L is linear, we then have L(\mathbf{x}) = 2 L(\mathbf{e}_1)+4 L(\mathbf{e}_2). Substitute the given values of L(\mathbf{e}_1) and L(\mathbf{e}_2)