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Composition of linear transformations

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Wize University Linear Algebra Textbook > Linear Transformations

Composition and Inverse

3 Activities

Wize University Linear Algebra Textbook > Linear Transformations

Special Linear Transformations in R2\reals^2

4 Activities

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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)

Find the matrix of the linear transformation that maps a vector u⃗\vec u in R2\mathbb{R}^2 to T(S(u⃗))T(S(\vec u)) .
More Composition and Inverse 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 and Inverse
Practice: Composition of Linear Transformations
Let T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3 be a linear transformation defined by T[xy]=[x+y3yx+2y]T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x+2y\end{bmatrix}.
Let S:R2→R2S:\reals^2 \to \reals^2 be the linear transformation induced by the matrix B=[1101]B= \left[ \begin{array}{rrr} 1 & 1 \\ 0 & 1 \\ \end{array} \right].
Matrix of a Linear Transformation
Consider the map T:R3→R3x⃗→proj(1,1,0)x⃗+2proj(1,0,−1)x⃗−proj(0,0,1)x⃗\begin{array}{l}T:\mathbb{R}^3\rightarrow\mathbb{R}^3\\\vec x\rightarrow proj_{(1,1,0)}\vec x+2proj_{(1,0,-1)}\vec x-proj_{(0,0,1)}\vec x\end{array}
a) Show that T is a linear transformation
b) Compute the matrix of T, and its inverse matrix.
Choose all the matrix products that are the identity matrix.
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]𝑩^𝟐=\begin{bmatrix} 0&1\\-1&0 \end{bmatrix}
Consider the matrix transformations SS and TT induced by the matrices A=[−1213]A=\begin{bmatrix}-1&2\\1&3 \end{bmatrix} and B=[122−1]B= \begin{bmatrix} 1&2\\2&-1 \end{bmatrix} , respectively. Then T∘ST\circ S is;
Let T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2 be a linear transformation with T[3−2]=[62]T\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix} and T[−54]=[−40]T\begin{bmatrix}-5\\4\end{bmatrix}=\begin{bmatrix}-4\\0\end{bmatrix}
(a) Find the matrix of TT, that is, find a matrix AA such that Tv⃗=Av⃗T\vec{v}=A\vec{v} for all v⃗∈R2\vec{v}\in\mathbb{R}^2.
(b) Is TT invertible? If so, find the matrix of T−1T^{-1}. If not, prove why not.
If TT and SS are linear transformations induced by the matrices
[−2024],[1−201]\begin{bmatrix}-2&0\\2&4\end{bmatrix},\quad \begin{bmatrix}1&-2\\0&1\end{bmatrix}
respectively, then find (S∘T)(x⃗)(S \circ T)(\vec{x}) where x⃗=[15]\vec{x}=\begin{bmatrix}1\\5\end{bmatrix}
Composition and Inverse
Example: Composition of Linear Transformations
Suppose TT and SS are linear transformations induced by the matrices A=[−2024], B=[10−21]A= \begin{bmatrix} -2&0\\ 2&4 \end{bmatrix}, \ B= \begin{bmatrix} 1&0\\ -2&1 \end{bmatrix}, respectively.
Part 1)
Given the linear transformation T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3 defined by T[xy]=[x+y3yx−2y]T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x-2y\end{bmatrix} , find the vector [xy]\begin{bmatrix}x\\y\end{bmatrix} such that T[xy]=[211]T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\1\\1\end{bmatrix}
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)
Let VV be the vector space {(x,y)  ∣  x,y∈R+}\{(x,y)\;|\;x,y\in\mathbb{R}^+\} with the operations:
(x1,y1)+(x2,y2)=(x1x2,y1y2)(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)
k⋅(x,y)=(xk,yk)k\cdot(x,y)=(x^k,y^k)
Let {e1,…,en}\{ e_1, \dots, e_n \} be a basis for an nn - dimensional vector space VV , and let L:V→WL : V\rightarrow W be a linear map, where WW is also a vector space. What conditions are necessary for {L(e1),…,L(en)}\{ L(e_1), \dots, L(e_n) \} to be a basis for WW ?
Practice: Composition and Matrix Inverse.
Suppose that T:Rn→RnT:\mathbb{R}^n\rightarrow\mathbb{R}^n is a linear transformation with inverse transformation S:Rn→RnS:\mathbb{R}^n\rightarrow\mathbb{R}^n , then, if AA is the matrix of TT , show that AA is invertible and A−1A^{-1} is the matrix of SS .
Given that T:R2→R2T:\mathbb{R}^2 \to \mathbb{R}^2is a linear transformation where:
T[30]=[6−9]T\begin{bmatrix}3\\0\end{bmatrix}=\begin{bmatrix}6\\-9\end{bmatrix}and T[01]=[11]T\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}1\\1\end{bmatrix}
Find T−1[2−1]T^{-1}\begin{bmatrix}2\\-1\end{bmatrix}
Composition of Transformations

Inverse of Transformations

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)
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