Linear Transformations: Why Are They Important?

Transformations

A transformation is a process by which an input vector 'x' is assigned an output L(x). In other words, a transformation is a function or a group of functions.

Linear Transformation

A linear transformation is a special kind of transformation that has two particular properties. If L is a linear transformation, and $\overrightarrow{v}$ and $\overrightarrow{w}$ are two vectors, then

1. L(v + w) = L(v) + L(w) (called Additivity or Distributivity)
2. L(c*v) = c*L(v) (called Homogeneity or Associativity)

When a transformation is additive and homogenous, we say it is a "linear transformation" In other words, if L is a linear transformation, and x = c1*v + c2*w, then:

$L\left(x\right)=c1\ast L\left(v\right)+c2\ast L\left(w\right)$

Let's look at a quick example

Linear Transformation: A Distortion of Space

Geometrically, a linear transformation can be interpreted as a distortion of the n-dimensional space in which the transformation is operating. Space can be distorted in several different ways. A linear transformation, however, is a special distortion with two distinct properties:

1. Lines remain lines
2. The origin remains fixed

The consequence of this distortion is such that lines that are parallel before the distortion, remain parallel after it.

Thus, a linear transformation can be visualised as a movement of the basis vectors to different locations.

Let's consider a linear transformation where the basis vectors move to the following points.

(Recall that the original basis vectors $\hat{i}$ and $\hat{j}$ are present at (1, 0) and (0, 1) respectively)

${\hat{i}}_{new}=\left[\begin{array}{c}2\\ 0\end{array}\right]{\hat{j}}_{new}=\left[\begin{array}{c}2\\ 3\end{array}\right]$

This means that $\hat{i}$ moves to (2, 0) and $\hat{j}$ moves to (2, 3) as part of the linear transformation.

Now, let's move on to matrices.