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a) If A is a rotation matrix for 30 degree CCW, find 𝑛 such that: 𝑨N = A-1 b…

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Wize University Linear Algebra Textbook > Linear Transformations

Composition and Inverse

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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}



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.
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