Matrix Multiplication

Matrix multiplication can be understood through three equivalent views:

1. Dot product view — each entry $(i,j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$
2. Column combination view — each column of $AB$ is a linear combination of columns of $A$, with coefficients from the corresponding column of $B$
3. Transformation view — $AB$ represents the composition of two linear transformations: first $B$, then $A$

Dimensions: if $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$. The inner dimensions must match.

Matrix multiplication is associative $(ABC = A(BC))$ but not commutative $(AB \neq BA$ in general$)$.

The transformation view is the most geometrically powerful: multiplying by a matrix rotates, scales, shears, or projects space. Composing transformations = multiplying matrices.

See also: Gaussian Elimination, Eigendecomposition, Rank and Null Space