Linear Transformations and Matrices. The linear transformations f :R m. −→ R n
are precisely the maps of the form. fA(x) = Ax, for x ∈ R m. , where A = [ a11 a21 ...
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
