Image of Symmetric Set under Linear Transformation is Symmetric

Theorem

Let $X$ and $Y$ be vector spaces over a subfield of $\C$.

Let $C$ be a symmetric subset of $X$.

Let $T : X \to Y$ be a linear transformation.

Then $\map T C$ is a symmetric subset of $Y$.

Proof

Let $y \in \map T C$.

Then there exists $x \in C$ such that $y = T x$.

Then from linearity we have $-y = \map T {-x}$.

Since $C$ is symmetric, we have $-x \in C$.

So $-y \in \map T C$.

So $\map T C$ is symmetric.

$\blacksquare$