Wize University Linear Algebra Textbook > Linear Transformations

Composition and Inverse

Composition and Inverse

Composition of Transformations

Suppose we have linear transformations $T:\mathbb{R}^k\to\mathbb{R}^n$ and $S:\mathbb{R}^n\to\mathbb{R}^m$ .

The composition of $S$ and $T$ is another linear transformation denoted $S \circ T: \reals^k \to \reals^m$ . It is defined as:

$\boxed{\quad \left(S\circ T\right)(\vec{x})=S(T(\vec{x})) \quad}$

If $A$ is the matrix inducing $T$ , and $B$ is the matrix inducing $S$ , then $BA$ is the matrix inducing $S\circ T$ :

$\begin{aligned} \left(S\circ T\right)(\vec{x}) &=S(T(\vec{x})) \\[0.3em] &=B(A\vec{x}) \\[0.3em] &= (BA)\vec{x} \end{aligned}$

Inverse of a Linear Transformation

Suppose $T:\mathbb{R}^n\to\mathbb{R}^n$ and $S:\mathbb{R}^n\to\mathbb{R}^n$ are linear transformations.

We say that $S$ and $T$ are inverses of one another if, for every $\vec{x}\in \reals^n$ :

$(S\circ T)(\vec{x})=\vec{x}$ and $(T\circ\ S)(\vec{x})=\vec{x}$

Matrix Form of the Inverse Transformation

If $A$ is the matrix inducing $T$ , then $T$ has an inverse transformation if and only if $A$ is invertible.

The inverse is unique and denoted by $T^{-1}:\mathbb{R}^n\to\mathbb{R}^n$ , and is induced by the matrix $A^{-1}$ .

Example: Composition of Linear Transformations

Suppose $T$ and $S$ are linear transformations induced by the matrices $A= \begin{bmatrix} -2&0\\ 2&4 \end{bmatrix}, \ B= \begin{bmatrix} 1&0\\ -2&1 \end{bmatrix}$ , respectively.

Part 1)

If $\vec{x}= \begin{bmatrix} 1\\ 5 \end{bmatrix}$ , find $(S \circ T)(\vec{x})$ .

$\begin{aligned} (S \circ T)(\vec x) &= BA\vec x\\[0.3em] &= \begin{bmatrix} 1&0\\ -2&1 \end{bmatrix} \begin{bmatrix} -2&0\\ 2&4 \end{bmatrix} \begin{bmatrix} 1\\ 5 \end{bmatrix}\\[1.5em] &= \colorTwo{ \begin{bmatrix} -2&0\\ 6&4 \end{bmatrix}} \begin{bmatrix} 1\\ 5 \end{bmatrix}\\[1.5em] &= \boxed{ \begin{bmatrix} -2\\ 26 \end{bmatrix}} \end{aligned}$

Note that $AB= \colorTwo{ \begin{bmatrix} -2&0\\ 6&4 \end{bmatrix}}$ is the matrix that induces $S \circ T$ .

Part 2)

Find the matrix inducing $T \circ S^{-1}$ .

Since $S$ is induced by the matrix $B$ , $S^{-1}$ must be induced by $B^{-1}$ .

$B= \begin{bmatrix} \colorOne1 & 0\\ -2 & \colorTwo1 \end{bmatrix} \quad\implies\quad B^{-1} \ =\ \begin{bmatrix} \colorTwo1& \colorFour{-}0\\ \colorFour{-}(-2) & \colorOne1 \end{bmatrix} \ =\ \begin{bmatrix} 1 & 0\\ 2 &1 \end{bmatrix}$

We know that for any vector $\vec x \in \reals^2$ :

$(T \circ S^{-1})(\vec x) = T(\, S^{-1}(\vec x)\, ) = T(B^{-1}\vec x) = AB^{-1}\vec x$

Therefore, the matrix inducing $T \circ S^{-1}$ is given by $AB^{-1}$ :

$AB^{-1} = \begin{bmatrix} -2&0\\ 2&4 \end{bmatrix} \begin{bmatrix} 1 & 0\\ 2 & 1 \end{bmatrix} = \boxed{ \begin{bmatrix} -2&0\\ 10&4 \end{bmatrix} }$

Practice: Composition of Linear Transformations

Let $T:\mathbb{R}^2\to\mathbb{R}^3$ be a linear transformation defined by $T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x+2y\end{bmatrix}$ .

Let $S:\reals^2 \to \reals^2$ be the linear transformation induced by the matrix $B= \left[ \begin{array}{rrr} 1 & 1 \\ 0 & 1 \\ \end{array} \right]$ .

Find the vector $\vec u = \begin{bmatrix} u\\ v \end{bmatrix}$ such that $(T\circ S^{-1})(\vec u) = \begin{bmatrix} 2\\ -3\\ 1 \end{bmatrix}$ .

$u=$$v=$

I don't know

Extra Practice

Inverse of Transformations

Composition of Transformations

Given that $T:\mathbb{R}^2 \to \mathbb{R}^2$ is a linear transformation where:

$T\begin{bmatrix}3\\0\end{bmatrix}=\begin{bmatrix}6\\-9\end{bmatrix}$ and $T\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}1\\1\end{bmatrix}$

Find $T^{-1}\begin{bmatrix}2\\-1\end{bmatrix}$

Practice: Composition and Matrix Inverse.

Suppose that $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a linear transformation with inverse transformation $S:\mathbb{R}^n\rightarrow\mathbb{R}^n$ , then, if $A$ is the matrix of $T$ , show that $A$ is invertible and $A^{-1}$ is the matrix of $S$ .

Let $\{ e_1, \dots, e_n \}$ be a basis for an $n$ - dimensional vector space $V$ , and let $L : V\rightarrow W$ be a linear map, where $W$ is also a vector space. What conditions are necessary for $\{ L(e_1), \dots, L(e_n) \}$ to be a basis for $W$ ?

Let $V$ be the vector space $\{(x,y)\;|\;x,y\in\mathbb{R}^+\}$ with the operations:

$(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)$

$k\cdot(x,y)=(x^k,y^k)$

Composition of linear transformations

Suppose that $A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]$

$T$ is the linear transformation induced by the matrix $A$ , and

$S$ is the linear transformation with matrix $B$ that rotates a vector by 60 degrees ( $\pi/3$ rad) counterclockwise around the point $(0,0)$

Given the linear transformation $T:\mathbb{R}^2\to\mathbb{R}^3$ defined by $T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x-2y\end{bmatrix}$ , find the vector $\begin{bmatrix}x\\y\end{bmatrix}$ such that $T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\1\\1\end{bmatrix}$

Composition and Inverse

Example: Composition of Linear Transformations

Suppose $T$ and $S$ are linear transformations induced by the matrices $A= \begin{bmatrix} -2&0\\ 2&4 \end{bmatrix}, \ B= \begin{bmatrix} 1&0\\ -2&1 \end{bmatrix}$ , respectively.

Part 1)

If $T$ and $S$ are linear transformations induced by the matrices

\begin{bmatrix}-2&0\\2&4\end{bmatrix},\quad \begin{bmatrix}1&-2\\0&1\end{bmatrix}

respectively, then find $(S \circ T)(\vec{x})$ where $\vec{x}=\begin{bmatrix}1\\5\end{bmatrix}$

Let $T:\mathbb{R}^2\to\mathbb{R}^2$ be a linear transformation with $T\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix}$ and $T\begin{bmatrix}-5\\4\end{bmatrix}=\begin{bmatrix}-4\\0\end{bmatrix}$

(a) Find the matrix of $T$ , that is, find a matrix $A$ such that $T\vec{v}=A\vec{v}$ for all $\vec{v}\in\mathbb{R}^2$ .

(b) Is $T$ invertible? If so, find the matrix of $T^{-1}$ . If not, prove why not.

Consider the matrix transformations $S$ and $T$ induced by the matrices $A=\begin{bmatrix}-1&2\\1&3 \end{bmatrix}$ and $B= \begin{bmatrix} 1&2\\2&-1 \end{bmatrix}$ , respectively. Then $T\circ S$ is;

a) If A is a rotation matrix for 30 degree CCW, find 𝑛 such that: ��N = A-1

b) Find two different rotation matrices B such that $𝑩^𝟐=\begin{bmatrix} 0&1\\-1&0 \end{bmatrix}$

Choose all the matrix products that are the identity matrix.

Matrix of a Linear Transformation

Consider the map $\begin{array}{l}T:\mathbb{R}^3\rightarrow\mathbb{R}^3\\\vec x\rightarrow proj_{(1,1,0)}\vec x+2proj_{(1,0,-1)}\vec x-proj_{(0,0,1)}\vec x\end{array}$

a) Show that T is a linear transformation

b) Compute the matrix of T, and its inverse matrix.

Composition and Inverse

Practice: Composition of Linear Transformations

Let $T:\mathbb{R}^2\to\mathbb{R}^3$ be a linear transformation defined by $T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x+2y\end{bmatrix}$ .

Let $S:\reals^2 \to \reals^2$ be the linear transformation induced by the matrix $B= \left[ \begin{array}{rrr} 1 & 1 \\ 0 & 1 \\ \end{array} \right]$ .

Composition of linear transformations

Suppose that $A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]$

$T$ is the linear transformation induced by the matrix $A$ , and

$S$ is the linear transformation with matrix $B$ that rotates a vector by 60 degrees ( $\pi/3$ rad) counterclockwise around the point $(0,0)$

Composition of linear transformations

Suppose that $A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]$

$T$ is the linear transformation induced by the matrix $A$ , and

$S$ is the linear transformation with matrix $B$ that rotates a vector by 60 degrees ( $\pi/3$ rad) counterclockwise around the point $(0,0)$

Composition of linear transformations

Suppose that $A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]$

$T$ is the linear transformation induced by the matrix $A$ , and

$S$ is the linear transformation with matrix $B$ that rotates a vector by 60 degrees ( $\pi/3$ rad) counterclockwise around the point $(0,0)$

Composition of linear transformations

Suppose that $A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]$

$T$ is the linear transformation induced by the matrix $A$ , and

$S$ is the linear transformation with matrix $B$ that rotates a vector by 60 degrees ( $\pi/3$ rad) counterclockwise around the point $(0,0)$

Composition of linear transformations

Suppose that $A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]$

$T$ is the linear transformation induced by the matrix $A$ , and

$S$ is the linear transformation with matrix $B$ that rotates a vector by 60 degrees ( $\pi/3$ rad) counterclockwise around the point $(0,0)$

Composition and Inverse

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