Question: When solving an exercise I have made the following step where $\alpha,\beta \in \mathbb{F}$ and $A,B,T\in M\_{n\times n}$

$$(\alpha A+\beta B)T=\alpha AT+\beta BT$$ 

Then I recalled the distributivity is not a property of a vector space, I know that left/right distributivity hold for matrices multiplication.
So there must be vector space with "Multiplication" that has no distributivity? Or there is just left/right distributivity in vector spaces?

Answer: 
The algebraic structure formed by matrices under addition and multiplication is called a ring. Rings have distributive properties.

If you want to explore structures with addition and multiplication but without distributive properties, perhaps check out [Ring without distributive law?](https://math.stackexchange.com/questions/120608/ring-without-distributive-law) and [Example of "ring" without the distributive property?](https://math.stackexchange.com/questions/513223/example-of-ring-without-the-distributive-property/513244)
Assuming dimensions work out, look at what feeding a vector on both sides does, you will see that it is the same, maybe setting $Tv=w$, another vector, will make things clearer
$$
(\alpha A+\beta B)Tv=(\alpha A+\beta B)w\stackrel{\text{linearity}}{=}
\alpha Aw+\beta Bw\\
=\alpha ATv+\beta BTv\implies (\alpha A+\beta B)T=\alpha AT+\beta BT
$$