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# Linear Transformations In the Plane

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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

# 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.

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