A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

18 Linear Algebra?

The standard basis of 4 × 4 grayscale images:

A linear combination (blend) of basis vectors compose an image:

Figure 2.8: Standard basis representation of grayscale images.

Linear spaces are associated with inner products. The inner

(or dot) product between vectors v and w is denoted by hv, wi,

as we already saw in Chapter 1. If the coordinates of these vectors

are given by v

1

, ··· , v

n

and w

1

, ··· , w

n

, respectively, their inner

product is:

hv, wi =

X

i

v

i

w

i

.

As discussed in Chapter 1, the dot product of two vectors gives us

useful information about angles and length, and generally about

the metric of the linear space.

Matrices as linear ...