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19.4F_Mid_Builder_$\tkcth{8.6.}$$\tkco{14}$_(8.6.1 without the general line)

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Wize University Linear Algebra Textbook > Linear Transformations

Special Linear Transformations in R2\reals^2

4 Activities

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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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.

  1. 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.
  2. Find a unit vector d^\dhat pointing in the same direction as d⃗\vd.
  3. Find T(d⃗)T(\vd) and T(d^)T(\dhat) by geometric reasoning (i.e. think about the effect of the transformation: no computation is required).
  4. Find any vector N⃗\vN perpendicular (i.e. normal) to the line y=15xy = \frac{1}{5}x. N.B. if the vector is perpendicular to the line, it will also be perpendicular to d⃗\vd.
  5. Find T(N⃗)T(\vN) by geometric reasoning (i.e. think about the effect of the transformation on this particular vector no computation is required).
  6. Use your results above to find T(ı^)=T(e^1)T(\ihat) = T(\ehat{1}) and T(ȷ^)=T(e^2)T(\jhat) = T(\ehat{2}).
  7. Find the standard matrix representation of the transformation TT.
  8. If possible, find the matrix representation of the inverse transformation T−1\minv{T}.
  9. Use geometric reasoning to find effect of applying this transformation twice in a row to the same vector, i.e. T ∘T(x⃗)=T(T(x))\bco{T}\, \circ \bcf{T(\bcb{\vx})} = \bco{T\big(} \bcf{T(\bcb{x})}\bco{\big)}.
  10. Verify your answer above by using the matrix representation of the transformation   ⁣M ⁣‾  \M on a general vector x⃗\vx. Do not change your previous answer to match it, if it was different.
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_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