On the book I am reading, $K_1$ group of a unital $C^*$-algebra $A$ is defined to be
$K_1(A)=\lim_{n\to\infty} GL(M_n(A))/GL(M_n(A))_0=\lim_{n\to\infty} U(M_n(A))/U(M_n(A))_0$,
where $A_0$ is the path component containing $1$ and $U(A)$ is the unitary group of $A$.
This definition is very easy to understand for me, but I never read the same definition anywhere else. Are the definitions all the same?