GEOMETRIC TRANSFORMATIONS - University of Texas at … - geometric transformation rotation calculator

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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


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