Solution
= Solution
The paths of $UX_t$ are continuous, and linear transformation preserves independence of increments. Moreover,
$$
U(X_t-X_s)\sim N\!\left(0,(t-s)UI_dU^T\right)
=N(0,(t-s)I_d)
$$
because $U$ is an <orthogonal matrix>. Thus all defining properties are preserved, proving the <orthogonal invariance of Brownian motion>:
$$
\boxed{(UX_t)_{t\geq0}\text{ is Brownian motion in }\mathbb R^d.}
$$