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Exercise 3.E.8

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Direction 1, $A$ is an affine subset:
Obviously we can define $A$ as,

A = v + U

where $v \in V$ and $U$ is a subspace of $V$ .
Let

x = v + u_{1}

and,

w = v + u_{2}

where $u_{1}, u_{2} \in U$ . Additionally define

u_{3} = λ (u_{1} - u_{2}) + u_{2}

Clearly, $u_{3} \in U$ So

\begin{matrix} \begin{aligned} v + u_{3} & \in A \\ λ (v - v) + v + λ (u_{2} - u_{2}) + u_{2} & \in A \\ λ (v + u_{1}) + (v + u_{2}) - λ (v + u_{2}) \in A \\ λ x + (1 - λ) w \in A \end{aligned} \end{matrix}

(2)

Direction 2, $λ v + (1 - λ) w \in A$
Now we are going to prove that $A$ is a subspace of $V$ and is thus an affine subset of $V$ .
Homogeneity: Let $v = - w$ . Thus,

\begin{matrix} \begin{aligned} λ v + (λ - 1) v & \in A \\ (2 λ - 1) v & \in A \end{aligned} \end{matrix}

(3)

Additivity: Let $λ = \frac{1}{2}$ .

\begin{matrix} \begin{aligned} \frac{1}{2} v + \frac{1}{2} w & \in A \\ v + w & \in A \end{aligned} \end{matrix}

(4)

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Arden_Shi

2025-06-02 23:49

Exercise 3.E.8

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