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Composition and Inverse

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Composition and Inverse
Composition of Transformations
Suppose we have linear transformations T:Rk→RnT:\mathbb{R}^k\to\mathbb{R}^nT:Rk→Rn and S:Rn→RmS:\mathbb{R}^n\to\mathbb{R}^mS:Rn→Rm.
The composition of SSS and TTT is another linear transformation denoted S∘T:Rk→RmS \circ T: \reals^k \to \reals^mS∘T:Rk→Rm. It is defined as:
 (S∘T)(x⃗)=S(T(x⃗))\boxed{\quad
\left(S\circ T\right)(\vec{x})=S(T(\vec{x}))
\quad}(S∘T)(x)=S(T(x))​
Wize Tip
To determine the right order in which to apply transformations, read the composition from right to left!
E.g. (U∘V)(x⃗)(U \circ V )(\vec x)(U∘V)(x) means find V(x⃗)\colorOne{V(\vec x)}V(x) first, then use that result to find U( V(x⃗) )U(\, \colorOne{V(\vec x)}\, )U(V(x)).

If AAA is the matrix inducing TTT, and BBB is the matrix inducing SSS, then BABABA is the matrix inducing S∘TS\circ TS∘T:
(S∘T)(x⃗)=S(T(x⃗))=B(Ax⃗)=(BA)x⃗\begin{aligned}
\left(S\circ T\right)(\vec{x})
&=S(T(\vec{x})) \\[0.3em]
&=B(A\vec{x}) \\[0.3em]
&= (BA)\vec{x}
\end{aligned}(S∘T)(x)​=S(T(x))=B(Ax)=(BA)x​
PAGE BREAK
Inverse of a Linear Transformation
Suppose T:Rn→RnT:\mathbb{R}^n\to\mathbb{R}^nT:Rn→Rn and S:Rn→RnS:\mathbb{R}^n\to\mathbb{R}^nS:Rn→Rn are linear transformations.
We say that SSS and TTT are inverses of one another if, for every x⃗∈Rn\vec{x}\in \reals^nx∈Rn:
(S∘T)(x⃗)=x⃗(S\circ T)(\vec{x})=\vec{x}(S∘T)(x)=x  and (T∘ S)(x⃗)=x⃗(T\circ\ S)(\vec{x})=\vec{x}(T∘ S)(x)=x

Wize Tip
Geometrically, the inverse transformation "undoes" the original transformation.
E.g. The inverse of a rotation by 20°20 \degree20° is a rotation by −20°-20 \degree−20°.

Matrix Form of the Inverse Transformation
If AAA is the matrix inducing TTT, then TTT has an inverse transformation if and only if AAA is invertible.
The inverse is unique and denoted by T−1:Rn→RnT^{-1}:\mathbb{R}^n\to\mathbb{R}^nT−1:Rn→Rn , and is induced by the matrix A−1A^{-1}A−1.

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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)
If x⃗=[15]\vec{x}=
\begin{bmatrix}
1\\
5
\end{bmatrix}x=[15​], find (S∘T)(x⃗)(S \circ T)(\vec{x})(S∘T)(x).

(S∘T)(x⃗)=BAx⃗=[10−21][−2024][15]=[−2064][15]=[−226]\begin{aligned}
(S \circ T)(\vec x) 
&= BA\vec x\\[0.3em]
&=
\begin{bmatrix}
1&0\\
-2&1
\end{bmatrix}

\begin{bmatrix}
-2&0\\
2&4
\end{bmatrix}

\begin{bmatrix}
1\\
5
\end{bmatrix}\\[1.5em]

&=
\colorTwo{
\begin{bmatrix}
-2&0\\
6&4
\end{bmatrix}}

\begin{bmatrix}
1\\
5
\end{bmatrix}\\[1.5em]

&=
\boxed{
\begin{bmatrix}
-2\\
26
\end{bmatrix}}

\end{aligned}(S∘T)(x)​=BAx=[1−2​01​][−22​04​][15​]=[−26​04​][15​]=[−226​]​​
Note that AB=[−2064]AB=
\colorTwo{
\begin{bmatrix}
-2&0\\
6&4
\end{bmatrix}}AB=[−26​04​] is the matrix that induces S∘TS \circ TS∘T.

Part 2)
Find the matrix inducing T∘S−1T \circ S^{-1}T∘S−1.

Since SSS is induced by the matrix BBB, S−1S^{-1}S−1 must be induced by B−1B^{-1}B−1.
B=[10−21]  ⟹  B−1 = [1−0−(−2)1] = [1021]B=
\begin{bmatrix}
\colorOne1 & 0\\
-2 & \colorTwo1
\end{bmatrix}

\quad\implies\quad

B^{-1}
\ =\ 
\begin{bmatrix}
\colorTwo1& \colorFour{-}0\\
\colorFour{-}(-2) & \colorOne1
\end{bmatrix}
\ =\ 
\begin{bmatrix}
1 & 0\\
2 &1
\end{bmatrix}B=[1−2​01​]⟹B−1 = [1−(−2)​−01​] = [12​01​]
We know that for any vector x⃗∈R2\vec x \in \reals^2x∈R2:
 (T∘S−1)(x⃗)=T( S−1(x⃗) )=T(B−1x⃗)=AB−1x⃗(T \circ S^{-1})(\vec x) = T(\, S^{-1}(\vec x)\, ) = T(B^{-1}\vec x) = AB^{-1}\vec x(T∘S−1)(x)=T(S−1(x))=T(B−1x)=AB−1x
Therefore, the matrix inducing T∘S−1T \circ S^{-1}T∘S−1 is given by AB−1AB^{-1}AB−1:
AB−1=[−2024][1021]=[−20104]AB^{-1} 
= 
\begin{bmatrix}
-2&0\\
2&4
\end{bmatrix}
\begin{bmatrix}
1 & 0\\
2 & 1
\end{bmatrix}
=
\boxed{
\begin{bmatrix}
-2&0\\
10&4
\end{bmatrix}
}
AB−1=[−22​04​][12​01​]=[−210​04​]​

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​].
Find the vector u⃗=[uv]\vec u =
\begin{bmatrix}
  u\\ v
\end{bmatrix}
u=[uv​] such that (T∘S−1)(u⃗)=[2−31](T\circ S^{-1})(\vec u)
=
\begin{bmatrix}
2\\
-3\\
1
\end{bmatrix}(T∘S−1)(u)=​2−31​​.

u=

v=

I don't know

Extra Practice

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 ?

Practice: Composition and Matrix Inverse.

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 .

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​]

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