1. **Problem statement:** Show that the reflection transformation $T:\mathbb{R}^2 \to \mathbb{R}^2$ about a line $\ell$ through the origin with slope $m \neq 0, \infty$ is linear, and find its standard matrix. 2. **Understanding reflection:** The line $\ell$ has slope $m$, so a direction vector for $\ell$ is $\mathbf{u} = \begin{bmatrix}1 \\ m\end{bmatrix}$. 3. **Reflection formula:** Reflection of a vector $\mathbf{v}$ about $\ell$ is given by $$T(\mathbf{v}) = 2\mathrm{proj}_{\mathbf{u}}(\mathbf{v}) - \mathbf{v}$$ where $\mathrm{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$. 4. **Check linearity:** Since projection and vector subtraction are linear operations, $T$ is linear. 5. **Find the matrix:** Calculate $T(\mathbf{v})$ for $\mathbf{v} = \begin{bmatrix}x \\ y\end{bmatrix}$: $$\mathbf{u} \cdot \mathbf{v} = x + my$$ $$\mathbf{u} \cdot \mathbf{u} = 1 + m^2$$ So, $$\mathrm{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{x + my}{1 + m^2} \begin{bmatrix}1 \\ m\end{bmatrix} = \begin{bmatrix} \frac{x + my}{1 + m^2} \\ \frac{m(x + my)}{1 + m^2} \end{bmatrix}$$ Then, $$T(\mathbf{v}) = 2 \begin{bmatrix} \frac{x + my}{1 + m^2} \\ \frac{m(x + my)}{1 + m^2} \end{bmatrix} - \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{2(x + my)}{1 + m^2} - x \\ \frac{2m(x + my)}{1 + m^2} - y \end{bmatrix}$$ 6. **Express as matrix multiplication:** $$T(\mathbf{v}) = \begin{bmatrix} \frac{2}{1 + m^2} - 1 & \frac{2m}{1 + m^2} \\ \frac{2m}{1 + m^2} & \frac{2m^2}{1 + m^2} - 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$ Simplify the entries: $$T = \begin{bmatrix} \frac{2}{1 + m^2} - 1 & \frac{2m}{1 + m^2} \\ \frac{2m}{1 + m^2} & \frac{2m^2}{1 + m^2} - 1 \end{bmatrix} = \begin{bmatrix} \frac{1 - m^2}{1 + m^2} & \frac{2m}{1 + m^2} \\ \frac{2m}{1 + m^2} & \frac{m^2 - 1}{1 + m^2} \end{bmatrix}$$ 7. **Final answer:** The reflection $T$ is linear with standard matrix $$\boxed{\begin{bmatrix} \frac{1 - m^2}{1 + m^2} & \frac{2m}{1 + m^2} \\ \frac{2m}{1 + m^2} & \frac{m^2 - 1}{1 + m^2} \end{bmatrix}}$$ This matrix reflects any vector in $\mathbb{R}^2$ about the line $\ell$ with slope $m$ through the origin.