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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?

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  • $\begingroup$ @Jack sorry my bad it is n by n matrices $\endgroup$
    – gbox
    Jan 3, 2018 at 15:28

2 Answers 2

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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? and Example of "ring" without the distributive property?

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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 $$ (\alpha A+\beta B)Tv=(\alpha A+\beta B)w\stackrel{\text{linearity}}{=} \alpha Aw+\beta Bw\\ =\alpha ATv+\beta BTv\implies (\alpha A+\beta B)T=\alpha AT+\beta BT $$

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