$$\frac{1}{2}x^THx+c^Tx + c_0$$
I have just formulated a problem as a quadratic optimization problem in two variables. My solution differs from the solution manual in the aspect that they have, only in front of the quadratic term, a coefficient $\frac{1}{2}$. Why is that there? Doesn't this change the equation?
For example:
I had two lines that are not parallel $L_1$, $L_2$ and I am supposed to find two points where the length is the smallest possible.
$$x = a + \alpha u$$ $$y = b + \beta v$$
where $a,b,u,v$ are known vectors in $R^3$ and $\alpha, \beta$ are in $R$
The final answer is
$$2x^THx+2c^Tx + c_0$$
The solution gives
$$x^THx+2c^Tx + c_0$$
Even if it is easier to differentiate, it just feels "illegal" to divide. Doesn't the equation change?