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If you think of matrix-vector multiplication geometrically as a linear transformation to a new coordinate system and matrix-matrix multiplication as the composition of two separate linear transformations, what does it mean to add two matrices together?

Would it make sense to think of it in terms of adding each basis vector separately to create a new set of basis vectors?

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    $\begingroup$ You can think of it as adding basis vectors. But beware that after the addition, the new set of vectors may not be independent any more. $\endgroup$ – P. Factor Nov 20 '18 at 18:23
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Linearity works both ways. That is, $$ (A+B)\vec{v} = A\vec{v} + B\vec{v}. $$ Thus, you can think of the linear transformation defined by $A+B$ as applied to the vector $\vec{v}$ as addition of the images under $A$ and $B$, separately, added together.

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