Page 238 has the key intuition. But to really understand it you will need some build up to understand linear functions. Matrices are simply a way to write dow

Page 238 has the key intuition. But to really understand it you will need some build up to understand linear functions. Matrices are simply a way to write down linear functions, and matrix multiplication is function composition. All of the algebraic properties of function composition are therefore true of matrices and vice versa. That intuition also explains why matrices come up so often. For example consider Calculus. The key idea behind the differential calculus is that if y is close to y0 then f(y) is approximately f(y0) + f'(y0) (y - y0). In the multivariable calculus the same thing is true. Except that f'(y0) is a linear function. Which means that when we write it down we have to write down a matrix.