Practice: Composition and Matrix Inverse.

Composition and Inverse

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Suppose that T:Rn→RnT:\mathbb{R}^n\rightarrow\mathbb{R}^nT:Rn→Rn is a linear transformation with inverse transformation S:Rn→RnS:\mathbb{R}^n\rightarrow\mathbb{R}^nS:Rn→Rn , then, if AAA is the matrix of TTT , show that AAA is invertible and A−1A^{-1}A−1 is the matrix of SSS .

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Composition of linear transformations

Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]A=[21​01​]
TTT is the linear transformation induced by the matrix AAA , and
SSSis the linear transformation with matrix BBB that rotates a vector by 60 degrees ( π/3\pi/3π/3 rad) counterclockwise around the point (0,0)(0,0)(0,0)

Composition of linear transformations

Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]A=[21​01​]
TTT is the linear transformation induced by the matrix AAA , and
SSSis the linear transformation with matrix BBB that rotates a vector by 60 degrees ( π/3\pi/3π/3 rad) counterclockwise around the point (0,0)(0,0)(0,0)

Composition of linear transformations

Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]A=[21​01​]
TTT is the linear transformation induced by the matrix AAA , and
SSSis the linear transformation with matrix BBB that rotates a vector by 60 degrees ( π/3\pi/3π/3 rad) counterclockwise around the point (0,0)(0,0)(0,0)

Composition of linear transformations

Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]A=[21​01​]
TTT is the linear transformation induced by the matrix AAA , and
SSSis the linear transformation with matrix BBB that rotates a vector by 60 degrees ( π/3\pi/3π/3 rad) counterclockwise around the point (0,0)(0,0)(0,0)

Composition of linear transformations

Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]A=[21​01​]
TTT is the linear transformation induced by the matrix AAA , and
SSSis the linear transformation with matrix BBB that rotates a vector by 60 degrees ( π/3\pi/3π/3 rad) counterclockwise around the point (0,0)(0,0)(0,0)

Composition and Inverse

Practice: Composition of Linear Transformations
Let T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3T:R2→R3 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}T[xy​]=​x+y3yx+2y​​.
Let S:R2→R2S:\reals^2 \to \reals^2S:R2→R2 be the linear transformation induced by the matrix B=[1101]B=
\left[
\begin{array}{rrr}
1 & 1 \\
0 & 1 \\
\end{array}
\right]B=[10​11​].

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}T:R3→R3x→proj(1,1,0)​x+2proj(1,0,−1)​x−proj(0,0,1)​x​
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}B2=[0−1​10​]

Consider the matrix transformations SSS and TTT induced by the matrices A=[−1213]A=\begin{bmatrix}-1&2\\1&3 \end{bmatrix}A=[−11​23​] and  B=[122−1]B= \begin{bmatrix} 1&2\\2&-1 \end{bmatrix}B=[12​2−1​] , respectively. Then T∘ST\circ ST∘S is;

Let T:R2→R2T:\mathbb{R}^2\to\mathbb{R}^2T:R2→R2 be a linear transformation with T[3−2]=[62]T\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix}T[3−2​]=[62​] and T[−54]=[−40]T\begin{bmatrix}-5\\4\end{bmatrix}=\begin{bmatrix}-4\\0\end{bmatrix}T[−54​]=[−40​]
(a) Find the matrix of TTT, that is, find a matrix AAA such that Tv⃗=Av⃗T\vec{v}=A\vec{v}Tv=Av for all v⃗∈R2\vec{v}\in\mathbb{R}^2v∈R2.
(b) Is TTT invertible? If so, find the matrix of T−1T^{-1}T−1. If not, prove why not.

If TTT and SSS 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}[−22​04​],[10​−21​]
respectively, then find (S∘T)(x⃗)(S \circ T)(\vec{x})(S∘T)(x) where x⃗=[15]\vec{x}=\begin{bmatrix}1\\5\end{bmatrix}x=[15​]

Composition and Inverse

Example: Composition of Linear Transformations
Suppose TTT and SSS 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}A=[−22​04​], B=[1−2​01​], respectively.
Part 1)

Given the linear transformation T:R2→R3T:\mathbb{R}^2\to\mathbb{R}^3T:R2→R3 defined by T[xy]=[x+y3yx−2y]T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x-2y\end{bmatrix}T[xy​]=​x+y3yx−2y​​ , find the vector [xy]\begin{bmatrix}x\\y\end{bmatrix}[xy​] such that T[xy]=[211]T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\1\\1\end{bmatrix}T[xy​]=​211​​

Composition of linear transformations

Suppose that A=[2011]A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]A=[21​01​]
TTT is the linear transformation induced by the matrix AAA , and
SSSis the linear transformation with matrix BBB that rotates a vector by 60 degrees ( π/3\pi/3π/3 rad) counterclockwise around the point (0,0)(0,0)(0,0)

Let VVV be the vector space {(x,y)  ∣  x,y∈R+}\{(x,y)\;|\;x,y\in\mathbb{R}^+\}{(x,y)∣x,y∈R+} with the operations:
(x1,y1)+(x2,y2)=(x1x2,y1y2)(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)(x1​,y1​)+(x2​,y2​)=(x1​x2​,y1​y2​)
k⋅(x,y)=(xk,yk)k\cdot(x,y)=(x^k,y^k)k⋅(x,y)=(xk,yk)

Let {e1,…,en}\{ e_1, \dots, e_n \}{e1​,…,en​} be a basis for an nnn - dimensional vector space VVV , and let L:V→WL : V\rightarrow WL:V→W be a linear map, where WWW is also a vector space. What conditions are necessary for {L(e1),…,L(en)}\{ L(e_1), \dots, L(e_n) \}{L(e1​),…,L(en​)} to be a basis for WWW ?

Given that T:R2→R2T:\mathbb{R}^2 \to \mathbb{R}^2T:R2→R2is a linear transformation where:
 T[30]=[6−9]T\begin{bmatrix}3\\0\end{bmatrix}=\begin{bmatrix}6\\-9\end{bmatrix}T[30​]=[6−9​]and T[01]=[11]T\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}1\\1\end{bmatrix}T[01​]=[11​]
Find T−1[2−1]T^{-1}\begin{bmatrix}2\\-1\end{bmatrix}T−1[2−1​]

Practice: Composition and Matrix Inverse.

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