Solution set of Linear Equations is an Affine Set

Last updated Nov 1, 2022

Let $V, W$ be a Vector Spaces over $\mathbb{R}$, let $T \in \mathcal{L}(V, W)$, and let $b \in W$ define the Linear Equation System $$T(x) = b.$$. Let $S = {x \in V : T(x) = b}$ be its Solution Set. Then $S$ is an Affine Set.

Let $\lambda \in \mathbb{R}$. Suppose $x_{1}, x_{2} \in S$. Then

$$\begin{align*} T(\lambda x_{1} + (1 - \lambda) x_{2}) &= \lambda T(x_{1}) + (1 - \lambda)T(x_{2})\\ &=\lambda b + (1 - \lambda) b\\ &=b \end{align*}$$ so $\lambda x_{1} + (1 - \lambda) x_{2} \in S$ and $S$ is an Affine Set. $\blacksquare$

TODO - Converse is true for $\mathbb{R}^{n}$ according to Boyd - Convex Optimization (Sect 2.1 pg 22, Example 2.1). Is this true in general?

• Real Numbers
• Linear Operator
• Linear Equation