Introduction to Linear Transformations

Vectors provide a very useful way to represent images in computer graphics and animation. These images are aptly referred to as vector graphics, and use $2-$dimensional arrangements of points which are connected by lines and curves to form polygons and other shapes. Vector graphics are found in PDF and ESP file formats $($as well as others$),$ and have many advantages over bitmap formats such as JPG and PNG which store images directly as a series of pixel values. For example, unlike JPG images, vector graphics can resized arbitrarily large without the images becoming pixelated. Furthermore, because images are represented as a series of vectors, they can be resized, rotated, reflected, and modified in other ways using techniques from linear algebra, such as linear transformations. In this lab we will explore linear transformations from a geometric viewpoint, and see simple applications to graphics.

The effect of magnifying vector graphics, which is clear, compared with magnifying a bitmap image, which is blurry.

Figure $1.$ The effect of magnifying vector graphics compared with magnifying a bitmap image. Image courtesy of Wikipedia$1.$

A transformation is a function, or rule, which assigns a vector in

to each vector in

$.$ For instance, a transformation

which maps vectors in

to vectors in

could be defined by

Thus, for instance,

We use the notation

to denote a transformations that accepts vectors in

as input, and returns vectors in

as output. We call the set

of all possible input vectors the domain of

$,$ and the set

where all of the output vectors lie the codomain of

$.$

A linear transformation is a special type of transformation which respects vector addition and scalar multiplication. More precisely, we say that a transformation

is a linear transformation from

into

if

for all vectors

and scalars

$.$

Every linear transformation

from

into

can be represented by an

matrix

$,$ called the standard matrix of

$,$ or equivalently the matrix representation of

$.$ For example, the linear transformation

given by

$(1)$Define a function transform$($x$)$ which takes as input a NumPy vector x in

$,$ performs the linear transformation

from $(1)$ above to x$,$ and returns the resulting vector as a NumPy vector.