Totally yes. The mechanistic illustrations of matrix multiplication, including the OP, are easy enough but don't help me with motivation or intuition. I always

Totally yes. The mechanistic illustrations of matrix multiplication, including the OP, are easy enough but don't help me with motivation or intuition. I always start with Ax, which is just a linear combination of the columns, like the first figure here https://eli.thegreenplace.net/2015/visualizing-matrix-multiplication-as-a-linear-combination/ https://eli.thegreenplace.net/2015/visualizing-matrix-multip... For those who want to make their own cheat sheet: Matrix $\times$ vector is linear combination of the columns: \begin{align} Ax &= \begin{pmatrix} a_{1,1} & a_{1,2}\\ a_{2,1} & a_{2,2}\\ a_{3,1} & a_{3,2} \end{pmatrix} \begin{pmatrix} x_1\\x_2 \end{pmatrix}\\ &= x_1\begin{pmatrix} a_{1,1}\\ a_{2,1}\\ a_{3,1}\\ \end{pmatrix} + x_2\begin{pmatrix} a_{1,2}\\ a_{2,2}\\ a_{3,2}\\ \end{pmatrix} \end{align} Intuitively, this example maps a point in $\mathbb{R}^2$ to $\mathbb{R}^3$ Another way to see $Ax$: Columns of $A$ form a basis, and $x$ is a coordinate vector in that basis.