High School
SAT
ACT
University
MCAT
LSAT
DAT

Composition of linear transformations

Related Topics

Wize University Linear Algebra Textbook > Linear Transformations

Special Linear Transformations in R2\reals^2

4 Activities

Wize University Linear Algebra Textbook > Linear Transformations

Composition and Inverse

3 Activities

checklist
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.
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 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.
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

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 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}
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