TABLE OF CONTENTS You already know the formula for matrix-vector multiplication. Nevertheless, the purpose of this first lecture is to describe a way of interpreting such products that may be less familiar. If b = Ax, then b is a linear combination of the columns of A.
Let x be an n-dimensional column vector and let A be an m n matrix ( m rows, n columns). Then the matrix-vector product b = Ax is the m-dimensional column vector defined as follows:
Here b i denotes the ith entry of b, a ij denotes the i, j entry of A ( ith row, jth column), and x j denotes the jth entry of x. For simplicity, we assume in all but a few lectures of this book that quantities such as these belong to 
, the field of complex numbers. The space of m-vectors is 
, and the space of m n matrices is 
.
The map x ? Ax is linear, which means that, for any 
and any 
,
Conversely, every linear map from 
can be expressed as multiplication by an m n matrix.
