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Special Linear Transformations

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Wize University Linear Algebra Textbook > Linear Transformations

Special Linear Transformations in R2\reals^2

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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.
More Special Linear Transformations in R2\reals^2 Questions:
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
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