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Multiplying a column vector
by a matrix
gives a new column vector
(possibly of a different dimension, depending on
the dimensions of ). If
is an
matrix, then
must be an
matrix, or an -dimensional column vector, and the product
will be
an
matrix, or an -dimensional column vector.
You can think of this multiplication as defining a *function*
from
-dimensional column vectors to -dimensional column vectors, given by

Functions of this kind are called
**linear**, and play an important role in mathematics. The main idea behind differentiation
is to produce linear approximations (like the tangent line approximation) to functions.
The reason that these functions are called linear is that they satisfy the same two
important linearity properties that linear differential operators satisfy: They
preserve addition and multiplication by constants.

**Example:** Suppose

Then
Similarly,

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*Peter Kostelec*

*2000-05-05*