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I know that matrix multiplication is not distributive but what about matrix-vector multiplication? If $A \in \mathbb{R^{m \times n}}$ and $\vec{x}+\delta \vec{x} \in \mathbb{R^{n \times 1}}$. Then can I write $A(\vec{x}+\delta \vec{x})$ as $A(\vec{x}) + A(\delta \vec{x})$? If yes, what is this property/law called?

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1 Answer

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Matrix-vector multiplication is a special case of matrix multiplication, which is distributive.

(In general, matrix multiplication is not commutative, but it is distributive.)

Your claim that $A(\vec{x}+\delta \vec{x})=A(\vec{x}) + A(\delta \vec{x})$ can also be seen as linearity.

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