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In linear algebra, I learned that the reason why matrix multiplication is defined in its way is because of composition of linear transformations. So, multiplication of two matrices refers to the composition of two linear transformations.

But, when we look at a matrix representation of systems of equations, $$\ Ax = b $$ where $A$ is a $m \times n$ matrix, $x$ is $n \times 1$ matrix of $n$ variables, and $b$ is $m \times 1$ matrix. We apply matrix multiplication here, but we don't refer this to the composition of linear transformations since it is a representation of systems of equations.

I'm confused about why we do matrix multiplication in the way it's defined even though it does not refer to composition of linear transformations.

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  • $\begingroup$ If you're interested in developing your intuition on this topic, I would recommend the video series youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab. The third and fourth videos specifically should help out a bit. $\endgroup$ Commented Jan 16, 2019 at 18:49

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If $A$ is the matrix of a linear map $L\colon V\longrightarrow W$ with respect to two bases $B$ and $B'$ and if the coordinates of a vector $v$ with respect to the bases $B$ are $a_1,\ldots,a_n$ (that is, if $v=a_1v_1+\cdots+a_nv_n$, with $B=\{v_1,\ldots,v_n\}$), then the coordinates of $L(v)$ with respect to the basis $B'$ are precisely the entries of the vector$$B.\begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix}$$So, multiplication of a matrix by a vector corresponds to computing the image of an element of $V$. And therefore, solving an equation $w=L(v)$ corresponds to a system of equations such as the one that you described.

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