  Using the Calder\'on-Zygmund decomposition, we give a novel and simple proof
that $L^2$ bounded dyadic shifts admit a domination by positive sparse forms
with linear growth in the complexity of the shift. Our estimate, coupled with
Hyt\"onen's dyadic representation theorem, upgrades to a positive sparse
domination of the class $\mathcal U$ of singular integrals satisfying the
assumptions of the classical $T(1)$-theorem of David and Journ\'e, with
logarithmic-Dini type smoothness of the integral kernel. Furthermore, our proof
extends rather easily to the $\mathbb R^n$-valued case, yielding as a corollary
the operator norm bound on the matrix weighted space $L^2(W; \mathbb R^n),$
  \[
  \left\|T\otimes \mathrm{Id}_{\mathbb R^n}\right\|_{L^2(W; \mathbb
R^n)\rightarrow L^2(W; \mathbb R^n)} \lesssim [W]_{A_2}^{\frac32}
  \] uniformly over $T\in \mathcal U$, which is the currently best known
dependence.
