Linear Transformations and Matrices
The linear transformations f : Rm −→ Rn are precisely
the maps of the form
for x ∈ Rm ,
fA (x) = Ax,
where A =
" a11
a12
..
.
a21 ... am1
a22 ... am2
.. . . ..
. .
.
a1n a2n ... amn
#
is an arbitrary n × m matrix.
Rescaling: The simplest types of linear transformations
are rescaling maps.
2
Consider
the
map
on
R
corresponding to the matrix
A = 20 03 .
That is,
x
y
7→
2 0
0 3
x
y
=
0-0
2x
3y
.
Shears:
The next simplest type of linear transformations are shears.
These are maps on R2 given by matrices like A =
1 α
0 1
,
which is a shear in the x–direction
For example, consider xy 7→ 10 21 xy = x+2y
.
y
b The
shear
b
b
b
b
b
1 0
1.5 1
The shear
b
b
12
01
Rescaling and Shear Transformations are not Euclidean
transformations in the sense that they generally change
distances and angles of two dimensional figures such as
triangles.
0-1
Rotations are linear transformations
Fix an angle θ ∈ [0, 2π) and consider the map
Rθ : R2 −→ R2
which is given by rotating through θ radians about the
Rθ (x) = (x′ , y ′ )
b
bx
θ
α
origin.
Write x = r cos α in polar coordinates.
Then Rθ (x) = r cos(α + θ), in polar coordinates.
Therefore, xy′ = Rθ (x) = rr cos(α+θ)
sin(α+θ)
α cos θ−sin α sin θ)
x cos θ − y sin θ
= r(cos
=
x sin θ + y cos θ
r(cos α sin θ+sin α cos θ)
x
θ − sin θ
= cos
y
sin θ cos θ
′
This shows that Rθ is a linear transformation.
0-2
= (x, y)
Reflections are linear transformations
Fix an angle θ ∈ [0, 2π) and consider Sθ : R2 −→ R2
which is given by reflecting through the line y = (tan θ)x.
′ ′
(x)
=
(x
,y )
S
θ
b
b
x = (x, y)
θ α
Write x = r cos α in polar coordinates.
Then Sθ (x) = r cos(2θ − α), in polar coordinates.
Therefore, xy′ = Sθ (x) = rr cos(2θ+α)
sin(2θ−α)
r(cos(2θ) cos α+sin(2θ) sin α)
= r(sin(2θ)
cos α−cos(2θ) sin α)
cos(2θ) + y sin(2θ)
= xx sin(2θ)
− y cos(2θ)
cos(2θ) sin(2θ)
= sin(2θ) − cos(2θ) xy
′
Hence, Sθ is a linear transformation.
0-3
Composing transformations
Suppose that f : V −→ W and g : W −→ X are linear
transformations.
Then g ◦ f : V −→ X is a linear transformation
To check we just need to compute:
Recall that (g ◦ f )(x) = g(f (x)).
So, if x, y ∈ V and r, s ∈ R then
(g ◦ f )(rx + sy) = g f (rx + sy)
= g rf (x)+ sf (y) = g rf (x) + g sf (y)
= rg(f (x))+sg(f (y))
r(g ◦ f )(x) + s(g ◦ f )(y)
Therefore, g ◦ f is a linear transformation.
0-4
Composing two reflections
By the last slides composing any two linear transformations gives a new linear transformation.
In particular, composing two reflections gives a new linear transformation. We work out what it is. Given two
angles α and β we compute Sα ◦ Sβ . For any xy ∈ R2
we have
= Sα
=
=
=
(Sα ◦ Sβ )
x
y
= Sα Sβ
!
cos(2β)
sin(2β)
sin(2β) − cos(2β)
cos(2α)
sin(2α)
sin(2α) − cos(2α)
x
y
x
y
cos(2β)
sin(2β)
sin(2β) − cos(2β)
!
x
y
cos(2α) cos(2β)+sin(2α) sin(2β) cos(2α) sin(2β)−sin(2α) cos(2β)
sin(2α) cos(2β)−cos(2α) cos(2β) sin(2α) sin(2β)+cos(2α) cos(2β)
cos 2(α−β) − sin 2(α−β)
sin 2(α−β)
sin 2(α−β)
x
y
Therefore, Sα ◦ Sβ = R2(α−β) .
In particular, composing two reflections gives a rotation!
0-5
x
y
Euclidean Transformations of R2
All Rotation and Reflection transformations of R2 are
Euclidean in the sense that they preserve distances and
angles.
Definition The transpose of an m × n matrix A − (aij )
is the n × m matrix (aji ) which is denoted by AT .
In simple terms, we swap the rows and the columns of a
matrix A to produce its transpose AT .
The rotation transformation Sθ and reflection transformation Rα we have described today both have the property
UUT = I
,
where U = Sθ or U = Rα . Any matrix U with this
property is known as an orthogonal matrix. The property
is equivalent to saying that the rows (or columns) of the
matrix are orthonormal vectors.
0-6