1. The problem is to understand why the norm is not linear on $\mathbb{R}^n$ and how the dot product helps introduce linearity. 2. The norm $\|\cdot\|$ on $\mathbb{R}^n$ is defined as $\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}$ for a vector $\mathbf{x} = (x_1, x_2, \ldots, x_n)$. 3. Linearity means two properties: additivity $f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y})$ and homogeneity $f(c\mathbf{x}) = c f(\mathbf{x})$ for scalar $c$. 4. The norm is not linear because it does not satisfy additivity: $\|\mathbf{x} + \mathbf{y}\| \neq \|\mathbf{x}\| + \|\mathbf{y}\|$ in general. 5. To introduce linearity, we use the dot product defined as $\mathbf{x} \cdot \mathbf{y} = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n$. 6. The dot product is linear in each argument: for fixed $\mathbf{y}$, $\mathbf{x} \mapsto \mathbf{x} \cdot \mathbf{y}$ is linear. 7. The norm can be expressed using the dot product as $\|\mathbf{x}\| = \sqrt{\mathbf{x} \cdot \mathbf{x}}$. 8. Thus, the dot product injects linearity into the discussion by providing a bilinear form from which the norm is derived. Final answer: The norm itself is not linear, but the dot product is a linear operation that helps define the norm and brings linear structure to $\mathbb{R}^n$.