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Let $\theta$ be a real number with continued fraction expansion $$\theta =\left[a_0,a_1,a_2,\cdots \right],$$ and let $$M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$ be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac{a\theta +b}{c\theta +d}=\left[a_0^{*},a_1^{*},a_2^{*},\cdots \right]$ also has bounded partial quotients. More precisely, if $a_j\le K$ for all sufficiently large $j$, then $a_j^{*}\le |det\left(M\right)|(K+2)$ for all sufficiently large $j$. We also give a weaker bound valid for all $a_j^{*}$ with $j\ge 1$. The proofs use the homogeneous Diophantine approximation constant $L_{\infty }\left(\theta \right)={lim\; sup}_{q\to \infty }{\left(q∥q^{\theta }∥\right)}^{-1}$. We show that $$\frac{1}{\left|det\left(M\right)\right|}{L}_{\infty }\left(\theta \right)\le L_{\infty }\left(\frac{a\theta +b}{c\theta +d}\right)\le \left|det\left(M\right)\right|L_{\infty }\left(\theta \right).$$