I have a property, in the context of Fourier Transform properties that says:

\[\delta \left ( a\cdot t \right ) = \frac{1}{a}\cdot \delta \left ( t \right )\]

and I need to apply it to the following expresison:

\[\delta \left ( \frac{\omega }{2} -1\right )\]

I say the result should be:

\[\begin{matrix} 1\rightarrow \delta \left ( \frac{\omega }{2} -1\right )\\ \\ 2\rightarrow \delta \left ( \frac{1}{2} \cdot \left ( \omega -2 \right )\right )\\ \\ 3\rightarrow 2\cdot \delta \left ( \omega -2 \right ) \end{matrix}

\]

where at:

\[2\]

I consider my 'a' to be:

\[\frac{1}{2}\]

and my 't' to be

\[\omega - 2\]

after looking to the left side of the property stated at the beginning.

My teacher wrote in the whiteboard that the result is:

\[\frac{1}{2}\delta \left ( \omega -2\right )\]

Am I wrong?