Geometric Transformations and Wallpaper Groups Vector and Matrix Algebra Wallpaper

advertisement
Wallpaper
Groups
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Geometric Transformations and Wallpaper
Groups
Vector and Matrix Algebra
Lance Drager
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas
2010 Math Camp
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Vectors
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Our setting is the Euclidean Plane
• A vector is a quantity that has a magnitude and a
direction.
Wallpaper
Groups
Vectors
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Our setting is the Euclidean Plane
• A vector is a quantity that has a magnitude and a
direction.
Wallpaper
Groups
Vector Operations
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• We’ll use letters like u, v , w , . . . to stand for vectors.
• 0 is the vector with magnitude zero. Just draw it as a
point.
• kuk denotes the magnitude of the vector u.
• A scalar is a quantity with a magnitude but no direction,
i.e., just a number.
• Two operations on vectors
• Scalar Multiplication
• Vector Addition.
Wallpaper
Groups
Scalar Multiplication
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Scalar multiplication sv .
• if s = 0, then sv is the zero vector.
• if s > 0, then sv has the same direction as v and
magnitude skv k.
• if s < 0, then sv points in the opposite direction to v and
has magnitude |s| times kv k.
• ksv k = |s| kv k.
Wallpaper
Groups
Visualizing Scalar Multiplication
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
2v
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
v
(−1/2)v
Wallpaper
Groups
Visualizing Vector Addition
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
u+v
v
u
Wallpaper
Groups
Visualizing Vector Subtraction
u − v = u + (−v )
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
u−v
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
v
−v
u
u−v
−v
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Lance Drager
Vectors in Coordinates
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• We can put a coordinate system on the plane.
y
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
(x, y)
y
x
x
Wallpaper
Groups
Lance Drager
Vectors in Coordinates
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• We can put a coordinate system on the plane.
y
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
(x, y)
y
x
• Distance formula: dist((0, 0), (x, y )) =
x
p
x2 + y2
Wallpaper
Groups
Lance Drager
Components of a Vector
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Coordinates give one-to-one correspondences
Vectors ↔ Points ↔ (u1 , u2 ) ∈ R2
y
P
u
(u1 , u2 )
x
Wallpaper
Groups
Lance Drager
Components of a Vector
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
• Coordinates give one-to-one correspondences
Vectors ↔ Points ↔ (u1 , u2 ) ∈ R2
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
y
P
u
(u1 , u2 )
x
• Write a vector as (u1 , u2 ). We call u1 and u2 the
q
components of u. kuk =
u12 + u22
Wallpaper
Groups
Scalar Multiplication in
Coordinates
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
(v1 , v2 )
y
v = su
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
(u1 , u2)
u
x
u2
v2
v2
=
=
kuk
kv k
skuk
v2 = su2
s(u1 , u2 ) = (su1 , su2 )
Wallpaper
Groups
Vector Addition in Coordinates
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
(w1 , w2 )
y
v1
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
(v1 , v2 )
w
(u1 , u2 )
v
u
u1
v1
x
w1 = u1 + v1
(u1 , u2 ) + (v1 , v2 ) = (u1 + v1 , u2 + v2 )
Wallpaper
Groups
Standard Basis Vectors
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
(u1 , u2 ) = u1 (1, 0) + u2 (0, 1) = u1 e1 + u2 e2
e1 = (1, 0),
e2 = (0, 1)
y
u2 e2
u
e2
e1
u1 e1 x
Wallpaper
Groups
A Nonstandard Basis
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
y
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
βv
w
αu
v
u
x
w = αu + βv
Wallpaper
Groups
Rules of Vector Algebra
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• (u + v ) + w = u + (v + w ). (Associative Law)
• u + v = v + u. (Commutative Law)
• 0 + v = v . (Additive Identity)
• v + (−v ) = 0. (Additive Inverse)
• s(u + v ) = su + sv . (Distributive Law)
• (s + t)u = su + tu. (Distributive Law)
• 1u = u.
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Lance Drager
Trig before Dot Product
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Give me an Intro to Trig
Wallpaper
Groups
Dot Product
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• The angle θ between two vectors.
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
v
θ
u
• Definition of dot product
u · v = kuk kv k cos(θ).
0 · v = 0.
• u · u = kuk2 .
• u · v = 0 ⇐⇒ u ⊥ v .
Wallpaper
Groups
Law of Cosines from Trig
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
c
b
θ
a
c 2 = a2 + b 2 − 2ab cos(θ).
Wallpaper
Groups
Formula for Dot Product
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
u−v
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
v
θ
u
ku − v k2 = kuk2 + kv k2 − 2kuk kv k cos(θ)
= kuk2 + kv k2 − 2u · v
2u · v = kuk2 + kv k2 − ku − v k2 .
Wallpaper
Groups
Lance Drager
Dot Product in Components
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
2u · v = kuk2 + kv k2 − ku − v k2
= k(u1 , u2 )k2 + k(v1 , v2 )k2 − k(u1 − v1 , u2 − v2 )k2
= u12 + u22 + v12 + v22 − [(u1 − v1 )2 + (u2 − v2 )2 ]
= u12 + u22 + v12 + v22 − (u1 − v1 )2 − (u2 − v2 )2
= u12 + u22 + v12 + v22 − (u12 − 2u1 v1 + v12 ) − (u22 − 2u2 v2 + v
= u12 + u22 + v12 + v22 − u12 + 2u1 v1 − v12 − u22 + 2u2 v2 − v22
= 2u1 v1 + 2u2 v2 .
u · v = u1 v1 + u2 v2 .
Wallpaper
Groups
Lance Drager
Properties of Dot Product
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Rules of Algebra for Dot Product
• u · u = kuk2 .
• u · v = v · u.
• (su) · v = s(u · v ) = u · (sv )
• u · (v + w ) = u · v + u · w .
• Compute the angle in terms of components
u·v
θ = arccos
.
kuk kv k
• Exercise: If u and v are orthogonal unit vectors (i.e.,
kuk = 1) and w = αu + βv , show α = w · u and β = w · v .
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Matrices
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• An m × n matrix is a rectangular array of numbers with m
rows and n columns.
• Example:
A=
−1 2 3
2 0 5
,
2 × 3.
• If the matrix is called A, aij denotes the entry in row i and
column j. For example, a23 = 5.
Wallpaper
Groups
Matrices
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• A general matrix

a11
 a21


A =  a31
 ..
 .
a12
a22
a32
..
.
a13
a23
a33
..
.
...
...
...
..
.

a1n
a2n 

a3n 
,
.. 
. 
m × n.
am1 am2 am3 . . . amn
• Matrix addition and scalar multiplication are defined
slotwise,
1
0
like vectors
3
7 4
1+7 3+4
8 7
+
=
=
5
10 1
0 + 10 5 + 1
10 6
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Matrix Multiplication 1
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• Size condition for multiplication:
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
(m × n) · (n × p) = m × p.
• (1 × n) · (n × 1) = 1 × 1
a1 a2 a3
 
b1
b2 


 
. . . an b3  = a1 b1 +a2 b2 +a3 b3 +· · ·+an bn .
 .. 
.
bn
Wallpaper
Groups
Matrix Multiplication 2
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• Entries of C = AB
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
cij = Rowi (A) Colj (B).
• Important Cases:
a11 a12 x1
a11 x1 + a12 x2
Ax =
=
.
a21 a22 x2
a21 x1 + a22 x2
a11 a12 b11 b12
AB =
a21 a22 b21 b22
a11 b11 + a12 b21 a11 b12 + a12 b22
=
.
a21 b11 + a22 b22 a21 b12 + a22 b22
Wallpaper
Groups
Lance Drager
Multiplication is not Commutative!
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• If AB and BA are the same size, they must both be n × n
for some n.
• Example
0
0
2
5
2 3
5
=
5 7
0
3 0 1
0
=
7 0 0
0
1
0
7
,
0
2
.
5
Wallpaper
Groups
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Ways of Looking at Multiplication
• Consider Ax
a11 a12 x1
a x + a12 x2
= 11 1
a21 a22 x2
a21 x1 + a22 x2
a11
a12
=
x +
x
a21 1
a22 2
Ax =
= x1 Col1 (A) + x2 Col2 (A).
• Ae1 = Col1 (A) and Ae2 = Col2 (A).
• Consider yB
b11 b12
y1 y1
= y1 b11 + y2 b21 y1 b12 + y2 b22
b21 b22
= y1 Row1 (B) + y2 Row2 (B).
• In terms of rows and columns
A[b1 | b2 ] = [Ab1 | Ab2 ],
a1
a B
B= 1 .
a2
a2 B
Wallpaper
Groups
Lance Drager
Rules of Matrix Algebra
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Rules of Algebra for Matrix Multiplication
• (AB)C = A(BC ). (Associative Law)
• A(B + C ) = AB + AC . (Left Distributive Law)
• (A + B)C = AC + BC . (Right Distributive Law)
• (sA)B = s(AB) = A(sB).
• A0 = 0 and 0B = 0.
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Invertible Matrices 1
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• In denotes the n × n identity matrix. It has ones on the
diagonal and zeros elsewhere.
1 0
,
I2 =
0 1


1 0 0
I3 = 0 1 0 .
0 0 1
• If A is m × n
Im A = A = AIn .
• A is invertible or nonsingular if there is a matrix B such
that
AB = BA = I
Wallpaper
Groups
Invertible Matrices 2
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Only square matrices can be invertible.
• B is unique, if it exists, and is denoted A−1 .
A−1 A = AA−1 = I .
• (A−1 )−1 = A.
• The Determinant of A =
a c
is defined by
b d
a c = ad − bc.
det(A) = b d
Wallpaper
Groups
Big Theorem on Invertible
Matrices
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Theorem
Let
a b
A=
c d
be a matrix. Then, the following conditions are equivalent.
1
A is invertible.
2
The only column vector x such that Ax = 0 is x = 0.
3
det(A) 6= 0.
Proof.
• (1) =⇒ (2). Multiply both sides of Ax = 0 by A−1 .
Wallpaper
Groups
Lance Drager
Big Theorem Proof Continued 1
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Proof Continued.
• (2) =⇒ (3). Instead, do: Not (3) =⇒ Not (2). Assume
ad − bc = 0. Of course, 0x = 0 for all x. Suppose A 6= 0.
Then at least one of the vectors
d
−c
,
is not zero. But,
−b
a
a c
d
ad − bc
=
= 0,
b d
−b
bd − bd
a c
−c
−ac + ac
=
= 0,
b d
a
−bc + ad
so (2) fails.
Wallpaper
Groups
Lance Drager
Big Theorem Proof Continued 2
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Proof Continued.
• (3) =⇒ (1). Just check that this formula works:
−1
A
1
d −c
.
=
det(A) −b a
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Exercises on Invertiblity
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Exercises
Here A and B are 2 × 2 matrices.
• If A and B are invertible, so is AB and (AB)−1 = B −1 A−1 .
• Show that if AB = I , then B = A−1 and if BA = I then
B = A−1 .
• A is invertible if and only if the equation Ax = b has a
unique solution x for every column vector b.
• Show that if det(A) = 0, one of the columns of A is
multiple of the other.
• Show by brute force computation (for 2 × 2 matrices) that
det(AB) = det(A) det(B).
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Linear Transformations
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Definition
A transformation T : R2 → R2 is linear if
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
T (u + v ) = T (u) + T (v )
T (su) = sT (u),
or, equivalently,
T (αu + βv ) = αT (u) + βT (v ).
Theorem
If A is a matrix, T (x) = Ax is linear.
Wallpaper
Groups
The Matrix of a Linear
Transformation
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Theorem
If T is linear, there is a unique matrix A so that T (x) = Ax. In
fact, A = [T (e1 ) | T (e2 )].
Proof.
T (x) = T (x1 e1 + x2 e2 )
= x1 T (e1 ) + x2 T (e2 )
x
= [T (e1 ) | T (e2 )] 1 .
x2
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
Rotation is Linear
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• Let T be the transformation given by rotation around the
origin by a fixed angle θ.
• We can see geometrically that T is linear.
Wallpaper
Groups
Rotations
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
y
u+v
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
v
u
x
Wallpaper
Groups
Rotations
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
y
y
u+v
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
T (u)
T (u + v)
v
T (v)
u
x
T (u + v ) = T (u) + T (v )
x
Wallpaper
Groups
Matrix of Rotation
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
• We need to find T (e1 ) and T (e2 ).
y
T (e2 )
T (e1 )
e2
θ
e1
x
Wallpaper
Groups
Find T (e1 )
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• We know T (e1 ) from trig:
y
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
T (e1 )
θ
e1
(cos(θ), sin(θ))
x
Wallpaper
Groups
Find T (e2 )
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
• T (e2 ) must be (− sin(θ), cos(θ)) or (sin(θ), − cos(θ)).
Check the signs.
y
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
T (e2 )
T (e1 )
e2
θ
e1
• T (e2 ) = (− sin(θ), cos(θ)).
x
Wallpaper
Groups
The Matrix of a Rotation
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Theorem
The matrix of rotation through angle θ is
cos(θ) − sin(θ)
R(θ) =
sin(θ)
cos(θ)
Theorem
From the geometry, we have
R(θ)R(φ) = R(θ + φ) = R(φ)R(θ)
Wallpaper
Groups
Outline
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
1 Vectors
Geometric Vectors
Vectors in Coordinates
The Dot Product
2 Matrices
What is a matrix?
Matrix Multiplication
Invertible Matrices
Exercises
Linear Transformations
Rotations
Exercises
Wallpaper
Groups
The Addition Laws
Lance Drager
Vectors
Geometric
Vectors
Vectors in
Coordinates
The Dot
Product
Matrices
What is a
matrix?
Matrix
Multiplication
Invertible
Matrices
Exercises
Linear
Transformations
Rotations
Exercises
Exercises
• From the geometry [R(θ)]−1 = R(−θ). Compute [R(θ)]−1
from our previous theorem. Compare entries in these
matrices to conclude cos(−θ) = cos(θ) and
sin(−θ) = − sin(θ).
• Compare the matrix entries in the equation
R(θ)R(φ) = R(θ + φ).
Then, replace φ by −φ and use the first part of the
problem.
You should get the Addition Laws for Sine and Cosine
cos(θ ± φ) = cos(θ) cos(φ) ∓ sin(θ) sin(φ),
sin(θ ± φ) = sin(θ) cos(φ) ± cos(θ) sin(φ).
Wallpaper
Groups
Trig Introduction
Lance Drager
Trig
Introduction
y
θ
x
• An angle determines a point on the unit circle.
Wallpaper
Groups
Lance Drager
Definition of Sine and Cosine
Trig
Introduction
y
(cos(θ), sin(θ))
θ
x
Wallpaper
Groups
The Basic Trig Identity
Lance Drager
Trig
Introduction
• A point (x, y ) is on the unit circle if and only if
p
x 2 + y 2 = 1 ⇐⇒ x 2 + y 2 = 1.
• (cos(θ), sin(θ)) is on the unit circle so
2 2
cos(θ) + sin(θ) = 1
cos2 (θ) + sin2 (θ) = 1
• Some people write sin(θ) = sin θ
Wallpaper
Groups
Non-Unit Circle
Lance Drager
Trig
Introduction
y
(r cos(θ), r sin(θ))
θ
(cos(θ), sin(θ))
x
Wallpaper
Groups
Triangles
Lance Drager
y
Trig
Introduction
(r cos(θ), r sin(θ))
hyp
θ
opp
adj
x
opp
r sin(θ)
=
= sin(θ)
hyp
r
adj
r cos(θ)
=
= cos(θ)
hyp
r
Back to Dot Product
Download