# Section 2.6 Linear Transformations

## 2.6A

Definition: A transformation (or mapping) $T$ is linear if

(1) $T(\vec{u}+\vec{v}) = T(\vec{u}) + T(\vec{v})$ for all $\vec{u},\vec{v}$ in the domain of $T$.

(2) $T(c\vec{u}) = cT(\vec{u})$ for all scalars $c$ and all $\vec{u}$ in the domain of $T$.

Fact: 1. Linear transformations preserve the operations of vector addition and scalar multiplication.

2. If $T$ is a linear transformation, then $T(\vec{0}) = \vec{0}$ and $T(c\vec{u}+d\vec{v}) = cT(\vec{u})+dT(\vec{v})$.

3. If a transformation satisfies $T(c\vec{u}+d\vec{v}) = cT(\vec{u})+dT(\vec{v})$ for all $\vec{u},\vec{v}$ in the domain of $T$ then it must be linear.

Theorem: If $T$ is a linear transformation then $T(c_{1}{v}_1 + \cdots + c_{p}\vec{v}_p) = c_{1}T(\vec{v}_1) + \cdots + c_{p}T(\vec{v}_p)$ for $\vec{v}_{1}, \cdots, \vec{v}_{p}$ in the domain of $T$. We call this equality superposition principle.

Example 1: Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation that maps $\vec{u} = \begin{bmatrix}1\\2\end{bmatrix}$ into $\begin{bmatrix}3\\4\end{bmatrix}$ and maps $\vec{v} = \begin{bmatrix}-1\\3\end{bmatrix}$ into $\begin{bmatrix}2\\-1\end{bmatrix}$. Use the fact that $T$ is linear to find the images under $T$ of $2\vec{u},-3\vec{v}$ and $2\vec{u}-3\vec{v}$.

Exercise 1: Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear transformation that maps $\vec{u} = \begin{bmatrix}1\\2\\-1\end{bmatrix}$ into $\begin{bmatrix}3\\5\\4\end{bmatrix}$ and maps $\vec{v} = \begin{bmatrix}-1\\4\\3\end{bmatrix}$ into $\begin{bmatrix}1\\-1\\-2\end{bmatrix}$. Use the fact that $T$ is linear to find the images under $T$ of $-(1/2)\vec{u},2\vec{v}$ and $-(1/2)\vec{u}+2\vec{v}$.

Fact: A matrix transformations is a linear transformation.

Definition: The standard basis of $\mathbb{R}^n$ is the columns set of $I_n, {\vec{e_1}, \cdots, \vec{e_n}}$.

Fact: Every vector $\vec{x} = \begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix}$ is a linear combination of the $\vec{e_i}$‘s and $\vec{x} = x_{1}\vec{e_1} + x_{2}\vec{e_2} + \cdots + x_{n}\vec{e_n}$.

Example 2: Let $\vec{e_1} = \begin{bmatrix}1\\0\end{bmatrix}, \vec{e_2} = \begin{bmatrix}0\\1\end{bmatrix}, \vec{y_1} = \begin{bmatrix}2\\-1\end{bmatrix}$ and $\vec{y_2} = \begin{bmatrix}1\\-1\end{bmatrix}$, and let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a linear transformation map $\vec{e_1}$ to $\vec{y_1}$ and $\vec{e_2}$ to $\vec{y_2}$. Find the image of $\vec{u} = \begin{bmatrix}-3\\4\end{bmatrix}$ and $\vec{x} = \begin{bmatrix}x_1\\x_2\end{bmatrix}$ under $T$.

Exercise 2: Let $\vec{e_1} = \begin{bmatrix}1\\0\end{bmatrix}, \vec{e_2} = \begin{bmatrix}0\\1\end{bmatrix}, \vec{y_1} = \begin{bmatrix}-2\\3\end{bmatrix}$ and $\vec{y_2} = \begin{bmatrix}4\\-1\end{bmatrix}$, and let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a linear transformation map $\vec{e_1}$ to $\vec{y_1}$ and $\vec{e_2}$ to $\vec{y_2}$. Find the image of $\vec{u} = \begin{bmatrix}6\\-5\end{bmatrix}$ and $\vec{x} = \begin{bmatrix}x_1\\x_2\end{bmatrix}$ under $T$.

Theorem: Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation. Then there exists a unique $m \times n$ matrix such that $T(\vec{x}) = A\vec{x}$ for all vectors $\vec{x}$ in $\mathbb{R}^n$.

In fact, $A = \begin{bmatrix}T(\vec{e_1}\cdots\vec{e_n})\end{bmatrix}$ where $\vec{e_i}$ is the i-th column of the $n \times n$ identity matrix. $A$ is called the standard matrix of $T$.

Proof: For any vector $\vec{x} = \begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix}$, we can write $\vec{x} = x_{1}\vec{e_{1}} + \cdots + x_{n}\vec{e_{n}}$ then by the fact that $T$ is a linear transformation, we have $T(\vec{x}) = x_{1}T(\vec{e_1}) + \cdots + x_{n}T(\vec{e_n}) = A\vec{x}$. The uniqueness is proved in the groupwork.

Theorem: Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a transformation. $T$ is linear if and only if it is a matrix transformation.

Theorem: Let $\mathbb{R}^k\overset{T}{\rightarrow}\mathbb{R}^n\overset{S}{\rightarrow}\mathbb{R}^m$ be linear transformations, and let $A$ and $B$ be the standard matrices of $S$ and $T$ respectively. Then $S \circ T$ is linear with standard matrix $AB$.

Example 3: Let $\mathbb{R}^k\overset{T}{\rightarrow}\mathbb{R}^n\overset{S}{\rightarrow}\mathbb{R}^m$ be linear transformation, and let $A=\begin{bmatrix}1 & 2\\0 & -1\\3 & 0\end{bmatrix}$ and $B=\begin{bmatrix}0 & -1 & 2 & 0\\1 & 2 & 0 & 1\end{bmatrix}$ be the standard matrices of $S$ and $T$ respectively. Find the standard matrix of $S \circ T$.

Exercise 3: Let $\mathbb{R}^k\overset{T}{\rightarrow}\mathbb{R}^n\overset{S}{\rightarrow}\mathbb{R}^m$ be linear transformation, and let $A=\begin{bmatrix}1 & 2 & 0\\0 & -1 & 1\end{bmatrix}$ and $B=\begin{bmatrix}0 & -1\\1 & 0\\1 & 2\end{bmatrix}$ be the standard matrices of $S$ and $T$ respectively. Find the standard matrix of $S \circ T$.

Example 4: Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation map $\begin{bmatrix}2\\1\end{bmatrix}$ to $\vec{y_{1}} = \begin{bmatrix}1\\-1\end{bmatrix}$ and $\begin{bmatrix}-1\\0\end{bmatrix}$ to $\vec{y_{2}} = \begin{bmatrix}2\\1\end{bmatrix}$. Find the image of $\vec{u} = \begin{bmatrix}-3\\2\end{bmatrix}$ under $T$.

Exercise 4: Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation map $\begin{bmatrix}1\\3\end{bmatrix}$ to $\vec{y_{1}} = \begin{bmatrix}2\\-1\end{bmatrix}$ and $\begin{bmatrix}-0\\2\end{bmatrix}$ to $\vec{y_{2}} = \begin{bmatrix}0\\1\end{bmatrix}$. Find the image of $\vec{u} = \begin{bmatrix}-3\\2\end{bmatrix}$ under $T$.

Group Work 1: Mark each statement True or False. Justify each answer.

a. A linear transformation is a special type of function.

b. If $A$ is a $3 \times 5$ matrix and $T$ is a transformation defined by $T(\vec{x}) = A\vec{x}$ then the domain of $T$ is $\mathbb{R}^5$.

c. If $A$ is an $m \times n$ matrix, then the range of the transformation $\vec{x} \rightarrow A\vec{x}$ is $\mathbb{R}^m$.

d. Every linear transformation is a matrix transformation.

e. A transformation $T$ is linear if and only if $T(c\vec{u} + d\vec{v}) = cT(\vec{u}) + dT(\vec{v})$ for all scalars $c, d$ and $\vec{u}, \vec{v}$ in the domain of $T$.

Group Work 2: Show the transformation $T$ defined by $T(x_{1}, x_{2}) = (x_{1}-3x_{2}, x_{1}-2, 3x_{1}-5x_{2})$ is not linear.

Group Work 3: Show the transformation $T$ defined by $T(x_{1}, x_{2}) = (x_{1}-3|x_{2}|, x_{1}-4x_{2})$ is not linear.

Group Work 4: Show the transformation $T$ defined by $T(x_{1}, x_{2}, x_{3}) = (x_{1}, 0, x_{3})$ is linear.

Group Work 5: Mark each statement True or False. Justify each answer.

a. The range of the transformation $\vec{x} \rightarrow A\vec{x}$ is the set of all linear combinations of the columns of $A$.

b. Every matrix transformation is a linear transformation.

c. A linear transformation preserves the operations of vector addition and
scalar multiplication.

d. A linear transformation, $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ always map the origin of $\mathbb{R}^n$ to the origin of $\mathbb{R}^m$.

e. Let $T$ and $S$ be linear transformations then $T \circ S = S \circ T$.

## 2.6B

Example 1: Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the transformation that rotates each point in $\mathbb{R}^2$ about the origin through an angle $\varphi$, with counterclockwise rotation for a positive angle. This transformation is a linear transformation. Find the matrix $A$ such that $T(\vec{x}) = A\vec{x}$.

Exercise 1: Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the transformation that rotates each point in $\mathbb{R}^2$ about the origin through an angle $\varphi$, with clockwise rotation for a positive angle. This transformation is a linear transformation. Find the matrix $A$ such that $T(\vec{x}) = A\vec{x}$.

Example 2: Find the standard matrix of a linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that first reflects points through the horizontal $x_{1}$-axis and then rotates points $-\frac{\pi}{2}$.

Exercise 2: Find the standard matrix of a linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that first reflects points through the horizontal $x_{1}$-axis and then reflects points through the line $x_{1}=x_{2}$.

Definition: 1. A map $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is called onto if for all $\vec{b}$ in $\mathbb{R}^m$ there is at least one
$\vec{x}$ in $\mathbb{R}^n$ such that $T(\vec{x}) = \vec{b}$.

2. A map $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is called one to one if $T(\vec{x}) = T(\vec{y})$ then $\vec{x} = \vec{y}$ for all $\vec{x}, \vec{y}$ in $\mathbb{R}^n$.

Example 3: Is the map defined by the matrix $A = \begin{bmatrix}2 & -2 & 3 & 1\\0 & 3 & 2 & 2\\0 & 0 & 0 & 4\end{bmatrix}$ one to one linear transformation?

Exercise 3: Is the map defined by the matrix $A = \begin{bmatrix}2 & -2 & -5 & 1\\0 & 3 & 4 & 1\\0 & 0 & 1 & 2\\0 & 0 & 0 & 0\end{bmatrix}$ one to one linear transformation?

Theorem: Let $\mathbb{R}^n\rightarrow\mathbb{R}^m$ be a linear transformations. $T$ is one to one if and only if $T(\vec{x}) = \vec{0}$ only has the trivial solution.

Theorem: Let $\mathbb{R}^n\rightarrow\mathbb{R}^m$ with standard matrix $A$. Then

(1) $T$ is onto if and only if $A$ has pivot positions in every row.

(2) $T$ is one to one if and only if $A$ has pivot position in every column.

Example 4: Let $T(x_{1}, x_{2}) = (x_{1} - x_{2}, 2x_{1} + 3x_{2}, 3x_{1} - 2x_{2})$. Show that $T$ is a one to one linear transformation. Is $T$ a onto transformation?

Example 4: Let $T(x_{1}, x_{2}) = (-x_{1} + x_{2}, 2x_{1} - x_{2}, -3x_{1} + 2x_{2})$. Show that $T$ is a one to one linear transformation. Is $T$ a onto transformation?

GroupWork 1: Mark each statement True or False. Justify each answer.

a. A linear transformation is completely determined by its effect on the
columns of the $n \times n$ matrix.

b. If $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ rotates vectors about the origin through an angle $\varphi$, then $T$ is a linear transformation.

c. When two linear transformations are performed one after another, the
combined effect may not always be a linear transformation.

d. A map $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is onto if every $\vec{x}$ in $\mathbb{R}^n$ maps onto some vector in $\mathbb{R}^m$.

e. If the standard matrix of a linear transformation is a $3 \times 2$ matrix then $T$ cannot be one to one.

Group Work 2: Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $S: \mathbb{R}^m \rightarrow \mathbb{R}^n$ be two linear transformations. Show the transformation from $\mathbb{R}^n$ to $\mathbb{R}^p$ defined by sending $\vec{x}$ in $\mathbb{R}^n$ to $S(T(\vec{x}))$ in $\mathbb{R}^p$ is a linear transformation.

Group Work 3: Mark each statement True or False. Justify each answer.

a. If the standard matrix of a linear transformation is $4 \times 3$ then the map is onto.

b. Every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ is a matrix transformation.

c. The columns of the standard matrix of a linear transformation $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is the images of the the columns of the $n \times n$ identity matrix under
$T$.

d. A linear transformation, $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is one to one if each vector in $\mathbb{R}^n$ maps to a unique vector in $\mathbb{R}^m$.

Group Work 4: A primitive society has three basic needs: food, shelter, and clothing. There are thus three industries in the societyâ€”the farming, housing, and garment industriesâ€”that produce these commodities. Each of these industries consumes a certain proportion of the total output of each commodity according to the following table.

Find the annual prices that each industry must charge for its income to equal its expenditures.