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Linear Transformations In the Plane

Linear Transformations In the Plane

Author: c o
Description:

To introduce the concept of a linear transformation
To understand the properties of linear transformations
To explore linear transformations by applying them to figures in two dimensional space

The learner is introduced to the concept of a linear transformation, which is then applied to transforming shapes in the plane.

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Tutorial

Linear Transformations

Background

Before diving into this packet, the learner should be familiar with the concept of a function, with matrix multiplication, and should be comfortable with the basic notion of vectors and vector spaces.

Definition

 

Example

The following examples show that multiplication by a matrix really is a linear transformation.  First, we show that multiplication by a constant can occur before or after applying the matrix:

 

Next we see that the map of the sum is the sum of the maps:

Transformations Of The XY-Plane

We can think of 2x2 matrices as transformations from R2 to R2.  That is, a 2x2 matrix M can be seen as a function that moves (x,y) points from one place to another.  This notion is employed extensively in software that manipulates images, in video games, and in many scientific applications.

The Kinds Of Transformation

Scaling can occur horizontally, vertically, or both.  The basic horizontal stretch is given by a matrix like this: , which moves a point (x,y) to the point (ax,y).  Similarly, the vertical stretch is given by a matrix like this: , which moves a a point (x,y) to the point (x,ay). To scale horizontally by a factor of a and vertically by a factor of b, apply the matrix .

 

Reflection moves a point from one side of a line to the other, and places it at a distance equal to the distance from the line to the original point..  A basic reflection across the x-axis is given by the matrix , which only changes the sign of the y-coordinate.  Likewise, reflection across the y-axis is given by the matrix , which changes the sign of the x-coordinate.  

 

Rotation moves a point circularly about the origin.  The matrix  rotates the plain counter-clockwise by an angle of a degrees.

 

Another way to alter points in a plane is to add a vector to them.  That is, if we are given the point (1,2), and we have a vector  <a,b>, then we can translate the point by the vector to the point (1+a,2+b).  This is not a linear transformation, but it does come up enough that you should know about it.

Examples Of Transformations Of The Plane

This is just a series of pictures showing how images can be manipulated with linear transformations.