Matrices move space

The single most useful reframe in linear algebra: a matrix is not a grid of numbers, it is a transformation of space that keeps grid lines parallel and evenly spaced and the origin fixed: rotation, scaling, shear, reflection or projection.

Read a 2×2 matrix column by column: the first column is where $\hat{i}$ lands, the second is where $\hat{j}$ lands. Every other vector follows, because it is a combination of $\hat{i}$ and $\hat{j}$. Drag the basis vectors in the simulation and watch the whole grid follow.

Matrix–vector multiplication $A\vec{x}$ is therefore 'apply transformation $A$ to $\vec{x}$'. A neural network layer $W\vec{x} + \vec{b}$ is exactly this, followed by a non-linearity.