Transformations from
One Vector Space to Another

We have seen many examples of procedures that take vectors in one vector space and map them or transform them into vectors in another vector space. For example, the rigid transformations that we studied in Flatland take vectors in R² and transform them into vectors in R² and in From Three Dimensions to Two Dimensions and Back and Looking at a Three Dimensional World with Two Dimensional Eyes we saw several ways in which vectors in R³ are mapped into vectors in R².

When a transformation, T maps vectors from a vector space U into vectors in a vector space V we say that U is the domain of T and that V is the range of T and we write.

T : U --> V

Example:

The procedures we developed in Analyzing a Periodic Function for taking a periodic function of period lambda and finding its Fourier Polynomials are examples of transformations whose domains are the vector spaces C[0, lambda].

Example:

The procedure we developed in From Three Dimensions to Two Dimensions and Back for finding the shadow cast by a point in R³ on the ground when it is illuminated by a light source is a transformation whose domain is R³ and whose range is R². Recall that we compute this transformation

as follows.

Suppose the light source is located at the point S = (s₁, s₂, s₃) and suppose that P = (p₁, p₂, p₃) is a point in R³. Then

where we need to solve the equations

t P + (1 - t) S = (x, y, 0)

s1 + t (p1 - s1) = x
s2 + t (p2 - s2) = y
s3 + t (p3 - s3) = 0

which leads to

       s3
t = -------  
    s3 - p3

and

x = s1 + t (p1 - s1)
y = s2 + t (p2 - s2)

We are interested in how these various transformations affect images. Use your CAS window to look at several different transformations and compare how they affect images. For example, look at the following.

• How do the various rigid transformations affect the square whose vertices are at
  (0, 0), (1, 0), (1, 1), and (0, 1)?

• How do the various ways that we map R³ into R² affect the z-axis? Be sure to look at the shadows cast by a light source. Does it make any difference where the light source is?

Definition:

We introduce the following terminology to describe some of the properties of transformations. We say that a transformation T : U --> V

preserves length

if for every vector u in U,

||T(u)|| = ||u||

is affine or preserves lines

if for any two vectors u and v in U and any real number t,

T(u + (1 - t) v) = t T(u) + (1 - t) T(v)

is linear

if for any two vectors u and v in U and any two real numbers a and b,

T(a u + b v) = a T(u) + b T(v)

Consider each of the transformations we have discussed. Which ones are linear? Which ones are affine? Which ones preserve length?