36. Matrix–Vector Product

A special case of matrix multiplication: a matrix times a single column vector.

For an matrix and a vector , the product has entries:

36.1. Two equivalent views

1. Row view (dot products): each entry is a dot product of a row of with .

2. Column view (linear combination): is a linear combination of ‘s columns, weighted by the entries of :

where is the -th column of .

This view explains why has a solution iff lies in the column space of .

36.2. Linear transformation view

Every matrix defines a linear transformation :

Conversely, every linear transformation between finite-dimensional spaces is a matrix–vector product (after fixing bases).

36.3. Linearity

applied to the zero vector returns the zero vector.