Our lecture defined a preference to be homothetic, if the following is true:
$$(x_1, x_2) \thicksim (y_1, y_2) \Leftrightarrow (kx_1, kx_2) \thicksim (ky_1, ky_2)$$
Cobb-Douglas preferences can be displayed as some utility function of the following form:
$$u(x_1, x_2) = x_1^a \cdot x_2^b$$ Therefore: $$(x_1, x_2) \thicksim (y_1, y_2) \\ \Leftrightarrow x_1^a \cdot x_2^b = y_1^a \cdot y_2^b \\ \Leftrightarrow k^ax_1^a \cdot k^bx_2^b = k^ay_1^a \cdot k^by_2^b \\ \Leftrightarrow (kx_1, kx_2) \thicksim (ky_1, ky_2)$$
With this argumentation the Cobb-Douglas preferences should be homothetic.
The wikipedia article about Homothetic preferences however defined a preference to be homothetic, if they can be represented by a utility function and the following is true:
$$ u(kx_1, kx_2) = k \cdot u(x_1, x_2)$$ And I am pretty sure, that this is not true for Cobb Douglas preferences:
$$ u(kx_1, kx_2) = (kx_1)^a (kx_2)^b = k^{a+b} x_1^a x_2^b \neq k \cdot u(x_1, x_2)$$
So what am I missing here? Are the definitions not equivalent? Did I calculate something wrong?