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$ V = M_{2,2} (\mathbb{R}), \forall M, N \in V : M \oplus N = MN, \forall M \in V, c \in \mathbb{R} : c \odot M = cM $

I know $\oplus$ and $\odot$ are closed, but a vectorspace also needs a neutral element, which would be $I_{2}$ in this case, and an inverse element which would be for $M^{-1}$ for $M \in V$. Now the inverse of M is not necessarily defined on a matrix, so would that be enough to disqualify this as a vectorspace? Would I need to take $V = Gl_2(\mathbb{R})$ for this to be a vectorspace? But then $\oplus$ might not be closed?

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up vote 7 down vote accepted

Yes, you do need additive inverses. But you run intro trouble even before that, because your $\oplus$ is not commutative. The addition in a vector space must be commutative.

Also, addition is required to distribute over scalar multiplication: $(cv)+(cw)=c(v+w)$ and $cd+dv=(c+d)v$, and neither of these hold for your operators.

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Vector addition must be commutative but matrix multiplication is not in general so this isn't a vector space.

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