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GEOMETRIC TRANSFORMATIONS

So far we've used vectors and matrices for the most part to write a system of linear equations in more concise

form as a vector or matrix equation. This often enabled us to interpret such properties of systems of equations as

unique or multiple solutions in different ways and to find these solutions systematically using row reduction, for

instance. Systems of linear equations still occur but more as a step on the way to understanding or interpreting

something rather than as something of importance in its own right.

Previously we associated an m ? n matrix A with a linear transformation

TA (x) = A x , TA : Rn Rm

using matrix-vector multiplication. And conversely, by Fundamental Theorem 1, each linear transformation

T : Rn Rm can be written as T = TA where A is the Standard Matrix. But frequently, a linear transformation T

is described in geometric terms or by some mathematical property, say, as rotation through of prescribed angle.

Let's see how this works for a number of geometric transformations T : R2 R2 . By the Fundamental Theorem all

that we need do is determine T (e1) and T (e2) where e1 and e2 correspond to the usual i = (1, 0) and j = (0, 1)

in the plane.

Rotation through counter-clockwise about the origin:

Since

T (e1) = T ([ 1 ]) = [ cos ] ,

0 sin

and

T (e2) = T ([ 0 ]) = [ - sin ] ,

1 cos

we get

A = [ cos - sin ] .

sin cos

Rotation through clockwise about the origin:

Since

T (e1) = T ([ 1 ]) = [ cos ] ,

0 - sin

and

T (e2) = T ([ 0 ]) = [ sin ] ,

1 cos

we get

cos sin ] .

A = [ - sin cos

Reflection through the x1-axis:

Since

T (e1) = T ([ 1 ]) = [ 1 ] ,

00

and

T (e2) = T ([ 0 ]) = [ 0 ] ,

1 -1

we get

A = [1 0 ].

0 -1

Get the idea? Now try to establish these yourself:

Reflection through the x2-axis: Reflection through the line x1 = x2:

A = [ -1 0 ] . A = [ 0 1 ] .

0 1 10

Reflection through the line x1 + x2 = 0: Reflection through the origin:

0 -1 ] . A = [ -1 0 ] .

A = [ -1 0 0 -1

Since TA TB = TAB for linear transformations, the standard matrix associated with compositions of geometric

transformations is just the matrix product AB.

Problem : find the Standard matrix for the linear

transformation T : R2 R2 which first

rotates points counter-clockwise about the origin

through /4,

and then

reflects points through the line x1 = x2.

Solution: the action of T is shown graphically to the

right. Now T is the composition TA TB of the matrix

transformation TB rotating R2 counter-clockwise

through /4 about the origin and the matrix

transformation TA reflecting R2 in the line x1 = x2

shown in purple, where Thus the Standard matrix for T = TA TB is

A = [ 0 1 ], B = [ cos /4 - sin /4 ]. [ 0 1 ]( 1 [ 1 -1 ]) = 1 [ 1 1 ] .

1 0 sin /4 cos /4 1 0 2 1 1 2 1 -1

One perhaps surprising consequence of this matrix/geometric approach to linear transformations is that familiar

trig identities can often be made completely natural and transparent. For suppose T defines rotation counter-

clockwise about the origin through , and T defines rotation counter-clockwise about the origin through . Then

Title: 05LinearTransformations

Author: John Gilbert

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