Section 2.2 Equations, Matrices, and Transformations Equations

2.2A

 

Definition: Consider the system of linear equations

 

$$\begin{array}{c}{a}_{11}{x}_1{a}_{12}{x}_2\cdots a_{1n}{x}_n=b_1\\ a_{21}{x}_1{a}_{22}{x}_2\cdots a_{2n}{x}_n=b_2\\ ⋮\\ a_{m1}{x}_1{a}_{m2}{x}_2\cdots a_{mn}{x}_n=b_m\end{array}$$

 

Such a system can be expressed in vector form or as a vector equation by using linear combinations of column vectors:

 

$$x_1\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ ⋮\\ a_{m1}\end{array}\right]x_2\left[\begin{array}{c}{a}_{12}\\ a_{22}\\ ⋮\\ a_{m2}\end{array}\right]\cdots x_n\left[\begin{array}{c}{a}_{1n}\\ a_{2n}\\ ⋮\\ a_{mn}\end{array}\right]=\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_m\end{array}\right]$$

 

Example 1: Write a vector equation that is equivalent to the given linear
system of equations.

 

$$\begin{array}{ccc}{x}_1{x}_3& =& 1\\ 2x_2-x_3& =& 2\\ 3x_1{x}_{2}3x_4& =& 3\end{array}$$

 

 

Exercise 1: Write a vector equation that is equivalent to the given linear system of equations.

 

$$\begin{array}{ccc}3x_2-x_3& =& 4\\ x_1-x_2{x}_3& =& -3\\ 2x_1{x}_2-2x_3& =& 2\end{array}$$

 

Definition: If $A$ is an $m\times n$ matrix, with column vectors, $\overrightarrow{v_1},\cdots ,\overrightarrow{v_n}$ in ${\mathbb{R}}^m$, and $\overrightarrow{x}$ is a vector in ${\mathbb{R}}^n$, then the product of $A$ and $\overrightarrow{x}$, denoted by $A\overrightarrow{x}$ in ${\mathbb{R}}^n$, is the linear combination of the columns of $A$ using the corresponding entries in $\overrightarrow{x}$ as weights; that is,

 

$$A\overrightarrow{x}=\left[\begin{array}{ccc}\overrightarrow{v_1}& \cdots & \overrightarrow{v_n}\end{array}\right]\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]=x_1\overrightarrow{v_1}{x}_2\overrightarrow{v_2}\cdots x_n\overrightarrow{v_n}$$

 

Note:
$A\overrightarrow{x}$ is defined only if the number of columns of $A$ equals the number of entries in $\overrightarrow{x}$.

 

 

Example 2: $A=\left[\begin{array}{ccc}2& 0& -1\\ 0& 1& 0\\ 1& 2& -2\\ -2& 1& 3\end{array}\right]$, $\overrightarrow{x}=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]$ find $A\overrightarrow{x}$.

 

Exercise 2: $A=\left[\begin{array}{cccc}2& 0& 1& -1\\ 0& 1& 2& 0\\ 1& 2& 3& -2\end{array}\right]$, $\overrightarrow{x}=\left[\begin{array}{c}2\\ 3\\ -1\\ 4\end{array}\right]$ find $A\overrightarrow{x}$.

 

Example 3: Write $-3\overrightarrow{u}2\overrightarrow{v}$ as a matrix times a vector where $\overrightarrow{u}=\left[\begin{array}{c}2\\ -1\end{array}\right]$ and
$\overrightarrow{v}=\left[\begin{array}{c}-3\\ 4\end{array}\right]$.

 

Exercise 3: Write $4\overrightarrow{u}(-2)\overrightarrow{v}3\overrightarrow{w}$ as a matrix times a vector where $\overrightarrow{u}=\left[\begin{array}{c}-1\\ -3\end{array}\right]$,
$\overrightarrow{v}=\left[\begin{array}{c}-2\\ 0\end{array}\right]$ and $\overrightarrow{w}=\left[\begin{array}{c}0\\ 2\end{array}\right]$.

 

Theorem: If $A$ is an $m\times n$ matrix, $\overrightarrow{u}$ and $\overrightarrow{v}$ are vectors in ${\mathbb{R}}^n$, and $c$ is a scalar, then

1. $A(\overrightarrow{u}\overrightarrow{v})=A\overrightarrow{u}A\overrightarrow{v}$

2. $A(c\overrightarrow{u})=cA\overrightarrow{u}.$

 

Definition: If $A$ is an $m\times n$ matrix, with column vectors, $\overrightarrow{v_1},\cdots ,\overrightarrow{v_n}$ in ${\mathbb{R}}^m$, and $\overrightarrow{x}$ is a vector in ${\mathbb{R}}^n$, and $\overrightarrow{b}$ is a vector in ${\mathbb{R}}^m$, then $A\overrightarrow{x}=\overrightarrow{b}$ is called a
matrix equation.

 

 

Theorem: If $A$ is an $m\times n$ matrix, with column vectors, $\overrightarrow{v_1},\cdots ,\overrightarrow{v_n}$ in ${\mathbb{R}}^n$, and $\overrightarrow{x}$ is a vector in ${\mathbb{R}}^n$, and $\overrightarrow{b}$ is a vector in ${\mathbb{R}}^m$, then the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ has the same solution set as the vector equation $x_1\overrightarrow{v_1}\cdots x_n\overrightarrow{v_n}=\overrightarrow{b}$, which, in turn, has the same solution set as the system of linear equations whose augmented matrix is $\left[\begin{array}{cccc}\overrightarrow{v_1}& \cdots & \overrightarrow{v_n}& \overrightarrow{b}\end{array}\right]$.

 

Fact: The equation $A\overrightarrow{x}=\overrightarrow{b}$ has a solution if and only if $\overrightarrow{b}$ is a linear combination of the columns of $A$.

 

Example 4: Solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ where $A=\left[\begin{array}{cc}2& 1\\ 1& -2\end{array}\right]$ and $\overrightarrow{b}=\left[\begin{array}{c}3\\ 4\end{array}\right]$.

Exercise 4: Solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ where $A=\left[\begin{array}{cc}1& 1\\ 3& -2\end{array}\right]$ and $\overrightarrow{b}=\left[\begin{array}{c}1\\ -1\end{array}\right]$.

 

Theorem: Let $A$ be an $m\times n$ matrix. Then the following statements are logically equivalent. That is, for a particular $A$, either they are all true statements or they are all false.

1. For each $\overrightarrow{b}$ in ${\mathbb{R}}^m$, the equation $A\overrightarrow{x}=\overrightarrow{b}$ has a solution

 

2. Each $\overrightarrow{b}$ in ${\mathbb{R}}^m$ is a linear combination of the columns of $A$.

 

3. $A$ has a pivot position in every row.

 

Example 5: Show each $\overrightarrow{b}$ in ${\mathbb{R}}^m$ is a linear combination of the columns of $A=\left[\begin{array}{ccc}1& -1& 2\\ 2& 0& 3\\ 0& 1& -2\end{array}\right]$

 

 

Exercise 5: Show each $\overrightarrow{b}$ in ${\mathbb{R}}^m$ is a linear combination of the columns of $A=\left[\begin{array}{ccc}0& -1& 2\\ 1& 3& 0\\ -2& 2& -2\end{array}\right]$

 

Definition: For each $n>2$, the identity matrix $I_n$ is the $n\times n$ matrix with 1’s on the main diagonal (upper left to lower right), and zeros elsewhere.

 

Fact: For each $n>2$, we have $I_n\overrightarrow{x}=\overrightarrow{x}$ for each n-vector $\overrightarrow{x}$ in ${\mathbb{R}}^n$.

 

Theorem: Let $A$ and $B$ be $m\times n$ matrices. If $A\overrightarrow{x}=B\overrightarrow{x}$ for all $\overrightarrow{x}$ in ${\mathbb{R}}^n$, then $A=B$.

 

 

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

a. The equation $A\overrightarrow{x}=\overrightarrow{b}$ is referred to as a vector equation.

 

b. A vector $\overrightarrow{b}$ is a linear combination of the columns of a matrix if and only if the equation $A\overrightarrow{x}=\overrightarrow{b}$ has at least one solution.

 

c. The equation $A\overrightarrow{x}=\overrightarrow{b}$ is consistent if the augmented matrix $\left[\begin{array}{cc}A& \overrightarrow{b}\end{array}\right]$ has a pivot position in every row.

 

d. The first entry in the product $A\overrightarrow{x}$ is a sum of products.

 

e. If $A$ is an $m\times n$ matrix and if the equation $A\overrightarrow{x}=\overrightarrow{b}$ is inconsistent for some $\overrightarrow{b}$ in ${\mathbb{R}}^m$ then $A$ cannot have a pivot position in every row.

 

f. The weight $c_1,\cdots ,c_p$ in a linear combination $c_1\overrightarrow{v_1}\cdots c_p\overrightarrow{v_p}$ cannot all be zero.

 

Group Work 2: Find the solution of $A\overrightarrow{x}=\overrightarrow{b}$ where

 

$A=\left[\begin{array}{cccc}1& 3& 3& 0\\ 0& 0& -1& 1\\ 0& 1& 2& 1\end{array}\right]$ and $\overrightarrow{b}=\left[\begin{array}{c}-2\\ 1\\ 1\end{array}\right]$

 

Group Work 3: In each case either show that the statement is true, or give
an example showing that it is false.

a. $\left[\begin{array}{c}3\\ 2\end{array}\right]$ is a linear combination of $\left[\begin{array}{c}1\\ 0\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$

 

b. $A\overrightarrow{x}$ has a zero entry, then $A$ has a row of zeros.

 

c. If $A\overrightarrow{x}=0$ where $\overrightarrow{x}\ne 0$, then $A=0$.

 

d. Every linear combination of vectors in ${\mathbb{R}}^n$ can be written in the form $A\overrightarrow{x}$.

 

e. If $A=\left[\begin{array}{ccc}\overrightarrow{a_1}& \overrightarrow{a_2}& \overrightarrow{a_3}\end{array}\right]$ in terms of its columns, and if $\overrightarrow{b}=3\overrightarrow{a_1}-2\overrightarrow{a_2}$, then the system $A\overrightarrow{x}=\overrightarrow{b}$ has a solution.

 

f. If $A=\left[\begin{array}{ccc}\overrightarrow{a_1}& \overrightarrow{a_2}& \overrightarrow{a_3}\end{array}\right]$ in terms of its columns, and if $A\overrightarrow{x}=\overrightarrow{b}$ has a solution then
$\overrightarrow{b}=s\overrightarrow{a_1}-t\overrightarrow{a_2}$ for some $s,t$.

 

g. If $A$ is an $m\times n$ matrix and $m<n$, then $A\overrightarrow{x}=\overrightarrow{b}$ has a solution for every column $\overrightarrow{b}$.

 

h. If $A\overrightarrow{x}=\overrightarrow{b}$ has a solution for some column $\overrightarrow{b}$, then it has a solution for every column $\overrightarrow{b}$.

2.2B

 

Definition: 1. A transformation (or function or mapping) $T$ from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ is a rule that assigns to each vector $\overrightarrow{x}$ in ${\mathbb{R}}^n$ to a vector $T(\overrightarrow{x})$ in ${\mathbb{R}}^m$.

 

2. The set ${\mathbb{R}}^n$ is called the domain of $T$ and ${\mathbb{R}}^m$ is called the codomain of $T$.

 

3. $T(\overrightarrow{x})$ is called the image of $\overrightarrow{x}$. The set {$T(\overrightarrow{x})$ for any $\overrightarrow{x}$ in ${\mathbb{R}}^n$} is called the range of $T$.

 

4. Given a $m\times n$ matrix $A$, we can define a transformation from
${\mathbb{R}}^n$ to ${\mathbb{R}}^M$ by defining $T(\overrightarrow{x})=A\overrightarrow{x}$. This kind of transformation is called a matrix transformation. $T$ is the matrix transformation induced by A.

 

 

Example 1: Let $A=\left[\begin{array}{cc}1& -2\\ -2& 4\\ 3& -6\end{array}\right]$, $\overrightarrow{b}=\left[\begin{array}{c}3\\ 4\end{array}\right]$, $\overrightarrow{c}=\left[\begin{array}{c}3\\ -6\\ 9\end{array}\right]$, and $\overrightarrow{d}=\left[\begin{array}{c}2\\ -6\\ 9\end{array}\right]$.

Let $T$ be the transformation defined by $T(\overrightarrow{x})=A\overrightarrow{x}$ from ${\mathbb{R}}^2$ to ${\mathbb{R}}^3$.

(a) Find $T(\overrightarrow{b})$.

(b) Find an $\overrightarrow{x}$ such that the image under $T$ is $\overrightarrow{c}$.

(c) Is there more than one $\overrightarrow{x}$ such that whose image under $T$ is $\overrightarrow{c}$?

(d) Is there an $\overrightarrow{x}$ such that its image under $T$ is $\overrightarrow{d}$?

 

 

Exercise 1: Let $A=\left[\begin{array}{cc}-1& 3\\ 2& -5\\ 2& -4\end{array}\right]$, $\overrightarrow{b}=\left[\begin{array}{c}5\\ 6\end{array}\right]$, $\overrightarrow{c}=\left[\begin{array}{c}-2\\ 5\\ 6\end{array}\right]$, and $\overrightarrow{d}=\left[\begin{array}{c}2\\ -6\\ 5\end{array}\right]$.

Let $T$ be the transformation defined by $T(\overrightarrow{x})=A\overrightarrow{x}$ from ${\mathbb{R}}^2$ to ${\mathbb{R}}^3$.

(a) Find $T(\overrightarrow{b})$.

(b) Find an $\overrightarrow{x}$ such that the image under $T$ is $\overrightarrow{c}$.

(c) Is there more than one $\overrightarrow{x}$ such that whose image under $T$ is $\overrightarrow{c}$?

(d) Is there an $\overrightarrow{x}$ such that its image under $T$ is $\overrightarrow{d}$?

 

Definition: Suppose $S:{\mathbb{R}}^n\to {\mathbb{R}}^m$ and $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ are transformations. Then $S=T$ if and only if $S(\overrightarrow{x})=T(\overrightarrow{x})$ for every $\overrightarrow{x}\in {\mathbb{R}}^n$.

 

Definition: 1. Given a matrix $A=\left[\begin{array}{cc}1& m\\ 0& 1\end{array}\right]$, where $m$ is a constant then the transformation $T$ defined by $T(\overrightarrow{x})=A\overrightarrow{x}$ is called a shear transformation. This kind of transformation send the vector $\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$ to $\left[\begin{array}{c}{x}_{1}m{x}_2\\ x_2\end{array}\right]$, i.e sending the point $(x_1,x_2)$ to the point $(x_{1}m{x}_2,x_2)$. $(x_1,x_2)$ is moving horizontally to the right if $m>0$ and $x_2>0$ or $m0$ and $x_{2}0$, otherwise is moving horizontally to the left.

 

 

2. Let $r$ be a scalar then the transformation $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ defined by $T(\overrightarrow{x})=r\overrightarrow{x}$ is called contraction when $0\le r1$ and dilation when $r>1$.

 

Example 2: Find the image of $\overrightarrow{u}=\left[\begin{array}{c}2\\ -3\end{array}\right]$ and $\overrightarrow{v}=\left[\begin{array}{c}-2\\ 3\end{array}\right]$ under the matrix transformation defined by $A=\left[\begin{array}{cc}1& 4\\ 0& 1\end{array}\right]$. Then graph the corresponding points on the plane.

 

 

Exercise 2: Find the image of $\overrightarrow{u}=\left[\begin{array}{c}-1\\ 2\end{array}\right]$ and $\overrightarrow{v}=\left[\begin{array}{c}2\\ -1\end{array}\right]$ under the matrix transformation $T$, defined by $A=\left[\begin{array}{cc}1& -3\\ 0& 1\end{array}\right]$. Describe geometrically what $T$ does to each vector.

 

Example 3: Find the image of $\overrightarrow{u}=\left[\begin{array}{c}1\\ -2\end{array}\right]$ and $\overrightarrow{v}=\left[\begin{array}{c}-3\\ 1\end{array}\right]$ under the matrix transformation $T$, defined by $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$. Then graph the corresponding points on the plane. Describe geometrically what $T$ does to each vector.

 

Exercise 3: Find the image of $\overrightarrow{u}=\left[\begin{array}{c}-1\\ 2\end{array}\right]$ and $\overrightarrow{v}=\left[\begin{array}{c}3\\ -5\end{array}\right]$ under the matrix transformation $T$, defined by $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$. Then graph the corresponding points on the plane. Describe geometrically what $T$ does to each vector.

 

Definition: The transformation $R_{\theta }:{\mathbb{R}}^2\to {\mathbb{R}}^2$ denotes counterclockwise rotation about the origin through an angle of $\theta$.

 

Example 4: The transformation $R_{\pi /2}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ denotes a counterclockwise rotation about the origin through an angle of $\pi /2$ radians. Find the matrix of $R_{\pi /2}$.

 

 

Exercise 4: The transformation $R_{\pi }:{\mathbb{R}}^2\to {\mathbb{R}}^2$ denotes a counterclockwise rotation about the origin through an angle of $\pi$ radians. Find the matrix of $R_{\pi }$.

 

Example 5: Let $T(\overrightarrow{x})=A\overrightarrow{x}$ where $\left[\begin{array}{ccc}1& 0& 5\\ 2& 3& 0\end{array}\right]$. Find a vector $\overrightarrow{x}$ such that $T(\overrightarrow{x})=\overrightarrow{b}=\left[\begin{array}{c}3\\ -1\end{array}\right]$.

 

 

Exercise 5: Let $T(\overrightarrow{x})=A\overrightarrow{x}$ where $\left[\begin{array}{ccc}0& 1& -3\\ -2& 2& 0\end{array}\right]$. Find a vector $\overrightarrow{x}$ such that $T(\overrightarrow{x})=\overrightarrow{b}=\left[\begin{array}{c}-2\\ 1\end{array}\right]$.

 

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 $T(\overrightarrow{x})=A\overrightarrow{x}$ then the domain of $T$ is ${\mathbb{R}}^5$.

 

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

 

d. The range of the transformation $\overrightarrow{x}\to A\overrightarrow{x}$ is the set of all linear combinations of the columns of $A$.

 

e. If $\overrightarrow{x_1}$ and $\overrightarrow{x_2}$ are solutions to $A\overrightarrow{x}=\overrightarrow{b}$, then $\overrightarrow{x_1}$ – $\overrightarrow{x_2}$ is a solution to $A\overrightarrow{x}=\overrightarrow{0}$.

 

f. Let $A=\left[\begin{array}{c}\overrightarrow{a_1}\overrightarrow{a_2}\overrightarrow{a_3}\end{array}\right]$ in terms of its columns. If $\overrightarrow{a_3}=s\overrightarrow{a_1}t\overrightarrow{a_2}$, then $A\overrightarrow{x}=0$, where $\overrightarrow{x}=\left[\begin{array}{c}s\\ t\\ -1\end{array}\right]$.

 

Group Work 2: Mark each statement True or False. Justify each answer.

Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a transformation. In each case show that $T$ is induced by a matrix and find the matrix.

a. $T$ is a reflection in the $y$ axis.

 

b. $T$ is a reflection in the line $y=x$.

 

c. $T$ is a reflection in the line $y=-x$.

 

d. $T$ is a clockwise rotation through $\pi /2$.

 

Group Work 3: In each case either show that the statement is true, or give
an example showing that it is false.

a. If a vector $\overrightarrow{b}$ is a linear combination of the columns of $A$, then the system $A\overrightarrow{x}=\overrightarrow{b}$ is consistent.

 

b. Let $\overrightarrow{x_1}$ and $\overrightarrow{x_2}$ be solutions to the homogeneous system $A\overrightarrow{x}=\overrightarrow{0}$ then $\overrightarrow{x_1}\overrightarrow{x_2}$ is a solution to $A\overrightarrow{x}=\overrightarrow{0}$.

 

c. Let $A$ and $B$ be matrices of the same size. If $\overrightarrow{x}$ is a solution to both the system $A\overrightarrow{x}=\overrightarrow{0}$ and the system $B\overrightarrow{x}=\overrightarrow{0}$, then $\overrightarrow{x}$ is a solution to the system $(AB)\overrightarrow{x}=\overrightarrow{0}$.

 

d. Let $A$ be $m\times n$ and let $\overrightarrow{x}$ be in ${\mathbb{R}}^n$. If $A\overrightarrow{x}$ has a zero entry then $A$ has a row of zeros.