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Rotation, Projection and Reflection Transformations
Related Topics
Wize University Linear Algebra Textbook > Linear Transformations
Special Linear Transformations in
R
2
\reals^2
R
2
4 Activities
Suppose that P is a projection, R is a rotation and Q is a reflection in 2D. Choose the statement below that is not true:
(a)
P
2
=
P
P^2=P
P
2
=
P
(b)
Q
2
=
I
Q^2=I
Q
2
=
I
(c)
R
T
=
R
−
1
R^T = R^{-1}
R
T
=
R
−
1
(d)
Q
=
Q
T
Q=Q^T
Q
=
Q
T
(e)
P
T
=
P
−
1
P^T = P^{-1}
P
T
=
P
−
1
I don't know
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More Special Linear Transformations in
R
2
\reals^2
R
2
Questions:
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Consider the matrix transformation
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
induced by the matrix
[
0
1
1
0
]
\begin{bmatrix}0&1\\1&0\end{bmatrix}
[
0
1
1
0
]
. Then
T
T
T
is:
Consider the matrix transformation
T
:
R
2
→
R
2
T:\mathbb{R}^2 \rightarrow \mathbb{R}^2
T
:
R
2
→
R
2
induced by the matrix
[
1
0
0
−
1
]
\left[ \begin{array}{cc} 1&0\\0&-1 \end{array} \right]
[
1
0
0
−
1
]
Then
T
T
T
is:
True or false: the matrix of
R
o
t
θ
Rot_{\theta}
R
o
t
θ
is indeed
[
cos
θ
−
sin
θ
sin
θ
cos
θ
]
\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}
[
cos
θ
sin
θ
−
sin
θ
cos
θ
]
Rotation, Projection and Reflection Transformations
Write down matrix of the rotation which transforms
T
(
[
0
1
]
)
=
[
−
1
0
]
T\left(\begin{bmatrix} 0\\1 \end{bmatrix}\right)=\begin{bmatrix} -1\\0 \end{bmatrix}
T
(
[
0
1
]
)
=
[
−
1
0
]
Special Linear Transformations
Example: Special Linear Transformations
Suppose
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
is a linear transformation that reflects a vector across the line
y
=
2
x
y = 2x
y
=
2
x
, then rotates the resulting vector through an angle of
60
°
60\degree
60°
counter-clockwise.
Find the matrix that induces
T
T
T
.
Consider the matrix transformation
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
induced by the matrix
[
−
1
0
0
1
]
\begin{bmatrix}-1&0\\0&1\end{bmatrix}
[
−
1
0
0
1
]
. Then
T
T
T
is:
Consider line L passing through the origin with direction [2,−4]. Find the matrix that represents
reflection across the line L
Let A be the matrix that rotates a 2D vector counterclockwise by 60, B be the matrix that reflects 2D vectors across the line
𝑦
=
3
𝑥
,
𝑦=\sqrt{3}𝑥,
y
=
3
x
,
and C be the projection to the line y=x. Which three of the following statements are true?
Consider line L passing through the origin with direction [2,−4]. Find the matrix that
represents reflection across the line L
a) If A is a rotation matrix for 30 degree CCW, find 𝑛 such that: 𝑨
n
= 𝑨
-1
b) Find two different rotation matrices B such that
𝑩
2
=
[
0
1
−
1
0
]
\text{ b) Find two different rotation matrices B such that }𝑩^2 =\begin{bmatrix} 0&1\\ -1&0 \end{bmatrix}
b) Find two different rotation matrices B such that
B
2
=
[
0
−
1
1
0
]
Consider the linear transformation T which takes vectors in R
2
, first rotates them by 30 degree counter-clockwise and then projects the result onto the direction
[
1
2
]
.
F
i
n
d
T
(
[
1
0
]
)
.
\begin{bmatrix} 1\\ 2 \end{bmatrix}.\ Find\ T\left(\begin{bmatrix} 1\\ 0 \end{bmatrix}\right).
[
1
2
]
.
F
in
d
T
(
[
1
0
]
)
.
Prove the matrix of
R
o
t
θ
Rot_{\theta}
R
o
t
θ
is indeed
[
cos
θ
−
sin
θ
sin
θ
cos
θ
]
\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}
[
cos
θ
sin
θ
−
sin
θ
cos
θ
]
True or false. Suppose
T
T
T
is a linear transformation with induced by
[
0
−
1
1
0
]
\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}
[
0
1
−
1
0
]
, then
T
T
T
is a rotation through the angle
90
°
90\degree
90°
counter-clockwise.
Special Linear Transformations
Example: Special Linear Transformations
Suppose
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
is a linear transformation that reflects a vector across the line
y
=
2
x
y = 2x
y
=
2
x
, then rotates the resulting vector through an angle of
60
°
60\degree
60°
counter-clockwise.
Find the matrix that induces
T
T
T
.
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
19.4F_Mid_Builder_$\tkcth{8.6.}$$\tkco{14}$_(8.6.1 without the general line)
Consider the transformation
T
:
R
2
→
R
2
T: \mathbb{R}^2 \to \mathbb{R}^2
T
:
R
2
→
R
2
that reflects a vector
x
⃗
\vx
x
across the line
y
=
1
5
x
y = \frac{1}{5}x
y
=
5
1
x
.
Find a vector parallel to the line. We call any vector that is a scalar multiple of the (parallel) direction of the line the
direction
vector of the line
d
⃗
\vd
d
.
Find a unit vector
d
^
\dhat
d
^
pointing in the same direction as
d
⃗
\vd
d
.
19.4F_Final_Builder_Ch_6.7_Curve_Fitting_$\tkco{eg1}(\tkco{a+b})$_$\key{Final}$_Builder_$\tkcth{6.7.}\tkcf{1}$_
Consider the transformation
T
:
R
2
→
R
2
T: \mathbb{R}^2 \to \mathbb{R}^2
T
:
R
2
→
R
2
that projects a vector
x
⃗
\vx
x
onto the line
y
=
5
x
y = 5x
y
=
5
x
.
Find a vector parallel to the line. We call any vector that is a scalar multiple of the (parallel) direction of the line the
direction
vector of the line
d
⃗
\vd
d
.
Find
T
(
d
⃗
)
T(\vd)
T
(
d
)
by geometric reasoning (
i.e.
think about the effect of the transformation: no computation is required).
Prove that for rotation matrix A in 2D, A
T
=A
-1
Prove that for rotation matrix A in 2D, A
T
=A
-1
There are two linear transformations of
R
2
\mathbb{R}^2
R
2
that map the square with vertices at (0,0); (1,0); (0,1);
and (1,1) to the square with vertices at (0,0); (1,1);(-1,1); and (0,2). Call them T
1
and T
2
.
Determine the matrices of T
1
and T
2
.
Special Linear Transformations
Practice: Special Linear Transformations
A linear transformation
T
:
R
2
→
R
2
T: \reals^2 \to \reals^2
T
:
R
2
→
R
2
is the result of the following transformations (in order):
first reflects across the line at an angle
π
4
rad
\dfrac{\pi}{4} \text{ rad}
4
π
rad
below the positive
x
x
x
-axis, then
Special Linear Transformations
Practice: Special Linear Transformations
Find the matrix inducing the linear transformation
S
:
R
2
→
R
2
S: \reals^2 \to \reals^2
S
:
R
2
→
R
2
that does the following:
first, rotates vectors counter-clockwise by
π
2
rad
\dfrac{\pi}{2} \text{ rad}
2
π
rad
, then
Common Transformations