High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Linear Algebra Textbook > Linear Transformations

Special Linear Transformations in R2\reals^2

Composition and InversePrevious Section
Eigenvalues and EigenvectorsNext Section
Popular Courses
Find My Course
0:00 / 0:00

Special Linear Transformations in R2\colorOne{\mathbb{R}^2}

Rotation

We define rotation in R2\reals^2 to be the linear transformation that rotates vectors by an angle θ\theta counter-clockwise.
The rotation matrix that induces this transformation is:
Rθ=[cos⁡θ−sin⁡θsin⁡θcos⁡θ]\boxed{\quad R_{\small \theta} = \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix} \quad}
Example
What is the result of rotating the vector [11]\begin{bmatrix} 1\\ 1\\ \end{bmatrix} counter-clockwise by π3 rad\dfrac{\pi}{3} \text{ rad}?
θ=π3  ⟹  Rθ=Rπ3=[cos⁡π3−sin⁡π3sin⁡π3cos⁡π3]=[1/2−3/23/21/2]\theta = \dfrac{\pi}{3} \quad \implies\quad R_{\small \theta} = R_{\normalsize\frac{\pi}{3}} = \begin{bmatrix} \cos\frac{\pi}{3} & -\sin\frac{\pi}{3}\\[0.5em] \sin\frac{\pi}{3} & \cos\frac{\pi}{3} \end{bmatrix} = \begin{bmatrix} 1/2 & -{\sqrt3}/{2}\\[0.5em] {\sqrt3}/{2} & {1}/{2} \end{bmatrix}
Then we just need to multiply: Rθ[11]=[1/2−3/23/21/2][11]=[1−321+32]R_{\small \theta} \begin{bmatrix} 1\\[0.5em] 1\\ \end{bmatrix} = \begin{bmatrix} 1/2 & -{\sqrt3}/{2}\\[0.5em] {\sqrt3}/{2} & {1}/{2} \end{bmatrix} \begin{bmatrix} 1\\[0.5em] 1\\ \end{bmatrix} = \boxed{ \begin{bmatrix} \dfrac{1-\sqrt3}{2}\\[1em] \dfrac{1+\sqrt3}{2}\\ \end{bmatrix} }
PAGE BREAK

Reflection

About Line y=mx\colorThree{y=mx}
First, we will look at the linear transformation that reflects vectors in R2\reals^2 across the line y=mxy=mx.
The reflection matrix that induces this transformation is:
Qm=11+m2[1−m22m2mm2−1]\boxed{\quad Q_m= \dfrac{1}{1+m^2}\begin{bmatrix} 1-m^2 & 2m\\[0.3em] 2m & m^2-1 \end{bmatrix} \quad}
About Line at Angle θ\colorThree{\theta}
We can also write this as a reflection in R2\reals^2 about the line set at an angle θ\theta counter-clockwise from the positive xx-axis.
This linear transformation is induced by the matrix:
Refθ=[cos⁡2θsin⁡2θsin⁡2θ−cos⁡2θ]\boxed{\quad \text{Ref}_{\small \theta} = \begin{bmatrix} \cos2\theta & \sin2\theta\\[0.3em] \sin2\theta & -\cos2\theta \end{bmatrix} \quad}
PAGE BREAK

Composition of Rotations and Reflections

The composition of two reflections or two rotations is a rotation.
two rotations:Rθ∘Rϕ=Rθ+ϕtwo reflections:Refθ∘Refϕ=R2(θ−ϕ)\begin{array}{lrcl} \text{two rotations:} &R_{\small\theta} \circ R_{\small\phi} &=& \boxed{R_{\small\theta+\phi}} \\[1em] \text{two reflections:} &\text{Ref}_{\small\theta} \circ \text{Ref}_{\small\phi} &=& \boxed{R_{\small 2(\theta-\phi)}} \end{array}
The composition of a reflection and a rotation is a reflection.
reflection, then rotation:Rθ∘Refϕ=Refϕ+θ/2rotation, then reflection:Refϕ∘Rθ=Refϕ−θ/2\begin{array}{lrcl} \text{reflection, then rotation:} &R_{\small\theta} \circ \text{Ref}_{\colorTwo{\small\phi}} &=& \boxed{\text{Ref}_{\small \colorTwo\phi + \theta/2}} \\[1em] \text{rotation, then reflection:} &\text{Ref}_{\colorTwo{\small\phi}} \circ \text{R}_{\small\theta} &=& \boxed{\text{Ref}_{\small \colorTwo\phi - \theta/2}} \end{array}

PAGE BREAK

Projection

We define projection in R2\reals^2 to be the linear transformation that projects a vector onto the line set at an θ\theta counter-clockwise from the positive xx-axis.
This linear transformation is induced by the matrix:
Projθ=12[1+cos⁡2θsin⁡2θsin⁡2θ1−cos⁡2θ]\boxed{\quad \text{Proj}_{\small \theta} = \dfrac{1}{2} \begin{bmatrix} 1+\cos2\theta & \sin2\theta\\[0.3em] \sin2\theta & 1-\cos2\theta \end{bmatrix} \quad}
0:00 / 0:00

Example: Special Linear Transformations

Suppose T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2 is a linear transformation that reflects a vector across the line y=2xy = 2x, then rotates the resulting vector through an angle of 60°60\degreecounter-clockwise.
Find the matrix that induces TT.
The matrix inducing the reflection is given by:
Q2=11+22[1−222(2)2(2)22−1]=15[−3443]Q_2 = \dfrac{1}{1+2^2} \begin{bmatrix} 1-2^2 & 2(2) \\[0.3em] 2(2) & 2^2-1 \end{bmatrix} = \dfrac{1}{5} \begin{bmatrix} -3 & 4 \\[0.3em] 4 & 3 \end{bmatrix}
The rotation matrix is given by:
R60°=[cos⁡(60°)−sin⁡(60°)sin⁡(60°)cos⁡(60°)]=[1/2−3/23/21/2]R_{\small 60\degree} = \begin{bmatrix} \cos(60\degree) & -\sin(60\degree) \\[0.3em] \sin(60\degree) & \cos(60\degree) \end{bmatrix} = \begin{bmatrix} 1/2 & -\sqrt{3}/2 \\[0.3em] \sqrt{3}/2 & 1/2 \end{bmatrix}
Then the matrix inducing TT is the product of these matrices (the rightmost happens first):
[1/2−3/23/21/2]⏞R60°  15[−3443]⏞Q2=15[−3/2−232−33/2−33/2+223+3/2]=110[−3−434−33−33+443+3]\begin{aligned} &\overbrace{ \begin{bmatrix} 1/2 & -\sqrt{3}/2 \\[0.5em] \sqrt{3}/2 & 1/2 \end{bmatrix}}^{\small R_{\small 60 \degree}} \ \ \overbrace{ \dfrac{1}{5} \begin{bmatrix} -3 & 4 \\[0.5em] 4 & 3 \end{bmatrix}}^{\small Q_{2}} \\[2em] = \dfrac{1}{5} &\begin{bmatrix} -3/2-2\sqrt{3} \quad & 2-3\sqrt{3}/2 \\[0.5em] -3\sqrt{3}/2+2 \quad & 2\sqrt{3}+3/2 \end{bmatrix}\\[2em] = \dfrac{1}{10} &\begin{bmatrix} -3-4\sqrt{3} \quad & 4-3\sqrt{3} \\[0.5em] -3\sqrt{3}+4 \quad & 4\sqrt{3}+3 \end{bmatrix} \end{aligned}

Practice: Special Linear Transformations

A linear transformation T:R2→R2T: \reals^2 \to \reals^2 is the result of the following transformations (in order):
  1. first reflects across the line at an angle π4 rad\dfrac{\pi}{4} \text{ rad} below the positive xx-axis, then
  2. rotates vectors 30°30 \degree counter-clockwise.
Find the matrix that induces this transformation.

Practice: Special Linear Transformations

Find the matrix inducing the linear transformation S:R2→R2S: \reals^2 \to \reals^2 that does the following:
  1. first, rotates vectors counter-clockwise by π2 rad\dfrac{\pi}{2} \text{ rad}, then
  2. projects the result onto the line −x+3⋅y=0-x+\sqrt3\cdot y=0 .
Extra Practice
Composition of linear transformations
Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
TT is the linear transformation induced by the matrix AA , and
SSis the linear transformation with matrix BB that rotates a vector by 60 degrees ( π/3\pi/3 rad) counterclockwise around the point (0,0)(0,0)
Composition of linear transformations
Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
TT is the linear transformation induced by the matrix AA , and
SSis the linear transformation with matrix BB that rotates a vector by 60 degrees ( π/3\pi/3 rad) counterclockwise around the point (0,0)(0,0)
Composition of linear transformations
Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
TT is the linear transformation induced by the matrix AA , and
SSis the linear transformation with matrix BB that rotates a vector by 60 degrees ( π/3\pi/3 rad) counterclockwise around the point (0,0)(0,0)
Composition of linear transformations
Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
TT is the linear transformation induced by the matrix AA , and
SSis the linear transformation with matrix BB that rotates a vector by 60 degrees ( π/3\pi/3 rad) counterclockwise around the point (0,0)(0,0)
Composition of linear transformations
Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
TT is the linear transformation induced by the matrix AA , and
SSis the linear transformation with matrix BB that rotates a vector by 60 degrees ( π/3\pi/3 rad) counterclockwise around the point (0,0)(0,0)
Consider the matrix transformation T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2 induced by the matrix [0110]\begin{bmatrix}0&1\\1&0\end{bmatrix} . Then TT is:
Consider the matrix transformation T:R2→R2T:\mathbb{R}^2 \rightarrow \mathbb{R}^2 induced by the matrix [100−1]\left[ \begin{array}{cc} 1&0\\0&-1 \end{array} \right]
Then TT is:
True or false: the matrix of RotθRot_{\theta} is indeed [cos⁡θ−sin⁡θsin⁡θcos⁡θ]\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}
Rotation, Projection and Reflection Transformations
Write down matrix of the rotation which transforms
T([01])=[−10]T\left(\begin{bmatrix} 0\\1 \end{bmatrix}\right)=\begin{bmatrix} -1\\0 \end{bmatrix}
Special Linear Transformations
Example: Special Linear Transformations
Suppose T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2is a linear transformation that reflects a vector across the line y=2xy = 2x, then rotates the resulting vector through an angle of 60°60\degreecounter-clockwise.
Find the matrix that induces TT.
Consider the matrix transformation T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2 induced by the matrix [−1001]\begin{bmatrix}-1&0\\0&1\end{bmatrix} . Then TT 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=3𝑥,𝑦=\sqrt{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: 𝑨n = 𝑨-1
 b) Find two different rotation matrices B such that 𝑩2=[01−10]\text{ b) Find two different rotation matrices B such that }𝑩^2 =\begin{bmatrix} 0&1\\ -1&0 \end{bmatrix}
Consider the linear transformation T which takes vectors in R2, first rotates them by 30 degree counter-clockwise and then projects the result onto the direction
[12]. Find T([10]).\begin{bmatrix} 1\\ 2 \end{bmatrix}.\ Find\ T\left(\begin{bmatrix} 1\\ 0 \end{bmatrix}\right).
Prove the matrix of RotθRot_{\theta} is indeed [cos⁡θ−sin⁡θsin⁡θcos⁡θ]\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}
True or false. Suppose TT is a linear transformation with induced by [0−110]\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix} , then TT is a rotation through the angle 90°90\degreecounter-clockwise.
Special Linear Transformations
Example: Special Linear Transformations
Suppose T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2 is a linear transformation that reflects a vector across the line y=2xy = 2x, then rotates the resulting vector through an angle of 60°60\degreecounter-clockwise.
Find the matrix that induces TT.
Composition of linear transformations
Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
TT is the linear transformation induced by the matrix AA , and
SSis the linear transformation with matrix BB that rotates a vector by 60 degrees ( π/3\pi/3 rad) counterclockwise around the point (0,0)(0,0)
19.4F_Mid_Builder_$\tkcth{8.6.}$$\tkco{14}$_(8.6.1 without the general line)
Consider the transformation T:R2→R2T: \mathbb{R}^2 \to \mathbb{R}^2 that reflects a vector x⃗\vx across the line y=15xy = \frac{1}{5}x.
Find a vector parallel to the line. We call any vector that is a scalar multiple of the (parallel) direction of the line the direction vector of the line d⃗\vd.
Find a unit vector d^\dhat pointing in the same direction as d⃗\vd.
19.4F_Final_Builder_Ch_6.7_Curve_Fitting_$\tkco{eg1}(\tkco{a+b})$_$\key{Final}$_Builder_$\tkcth{6.7.}\tkcf{1}$_
Consider the transformation T:R2→R2T: \mathbb{R}^2 \to \mathbb{R}^2 that projects a vector x⃗\vx onto the line y=5xy = 5x.
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 d⃗\vd.
Find T(d⃗)T(\vd) by geometric reasoning (i.e. think about the effect of the transformation: no computation is required).
Prove that for rotation matrix A in 2D, AT=A-1
Prove that for rotation matrix A in 2D, AT=A-1
There are two linear transformations of R2\mathbb{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 T1 and T2.
Determine the matrices of T1 and T2.
Special Linear Transformations
Practice: Special Linear Transformations
A linear transformation T:R2→R2T: \reals^2 \to \reals^2 is the result of the following transformations (in order):
first reflects across the line at an angle π4 rad\dfrac{\pi}{4} \text{ rad} below the positive xx-axis, then
Special Linear Transformations
Practice: Special Linear Transformations
Find the matrix inducing the linear transformation S:R2→R2S: \reals^2 \to \reals^2 that does the following:
first, rotates vectors counter-clockwise by π2 rad\dfrac{\pi}{2} \text{ rad}, then
Rotation, Projection and Reflection Transformations
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:
Common Transformations