Mathematics for Data Science II Quiz 2: 23 November 2025 (September 2025 term)

Which of the following is an affine subspace but not a vector subspace?

Consider $\mathbb{R}^3$ with the dot product and define two functions $f : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}$ and $g : \mathbb{R}^3 \to \mathbb{R}^{\geq 0}$, where $\mathbb{R}^{\geq 0} := \{x \in \mathbb{R} \mid x \geq 0\}$, by $f(v,w) = (5-k)v \cdot w$ and $g(v) = (6+k)\left(\sqrt{v_1^2 + v_2^2 + v_3^2}\right)$, where $v = (v_1, v_2, v_3)$.

Let $S_1$ denote the set of all values of $k \in \mathbb{Z}$ such that $f$ is an inner product on $\mathbb{R}^3$ and $S_2$ denote the set of all values of $k \in \mathbb{Z}$ such that $g$ defines a norm on $\mathbb{R}^3$.

Based on the above data, answer the given subquestions.

Consider $\mathbb{R}^3$ with the dot product and define two functions $f : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}$ and $g : \mathbb{R}^3 \to \mathbb{R}^{\geq 0}$, where $\mathbb{R}^{\geq 0} := \{x \in \mathbb{R} \mid x \geq 0\}$, by $f(v,w) = (5-k)v \cdot w$ and $g(v) = (6+k)\left(\sqrt{v_1^2 + v_2^2 + v_3^2}\right)$, where $v = (v_1, v_2, v_3)$.

Let $S_1$ denote the set of all values of $k \in \mathbb{Z}$ such that $f$ is an inner product on $\mathbb{R}^3$ and $S_2$ denote the set of all values of $k \in \mathbb{Z}$ such that $g$ defines a norm on $\mathbb{R}^3$.

Based on the above data, answer the given subquestions.