# Section 2.2 Equations, Matrices, and Transformations Equations

## 2.2A

Definition: Consider the system of linear equations

$\begin{array}{cccc} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} = b_{1}\\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} = b_{2}\\ \vdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} = b_{m}\\ \end{array}$

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

$x_{1}\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix} + x_{2}\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix} + \cdots + x_{n}\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix} = \begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}$

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.

$\begin{array}{ccc} 3x_{2}-x_{3} & = & 4\\ x_{1}-x_{2}+x_{3} & = & -3\\ 2x_{1}+x_{2}-2x_{4} & = & 2 \end{array}$

Definition: If $A$ is an $m \times n$ matrix, with column vectors,
$\vec{v_{1}}, \cdots, \vec{v_{n}}$ in $\mathbb{R}^m$, and $\vec{x}$ is a vector in $\mathbb{R}^n$, then the product of $A$ and $\vec{x}$, denoted by $A\vec{x}$ in $\mathbb{R}^n$, is the linear combination of the columns of $A$ using the corresponding entries in $\vec{x}$ as weights; that is,

$A\vec{x} = \begin{bmatrix}\vec{v_{1}} & \cdots & \vec{v_{n}}\end{bmatrix} \begin{bmatrix}x_{1} \\\vdots\\ x_{n} \end{bmatrix} = x_{1}\vec{v_{1}}+x_{2}\vec{v_{2}}+\cdots+x_{n}\vec{v_{n}}$

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

Example 2: $A = \begin{bmatrix} 2 & 0 & -1\\ 0 & 1 & 0\\ 1 & 2 & -2\\ -2 & 1 & 3 \end{bmatrix}$, $\vec{x} = \begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$ find $A\vec{x}$.

Exercise 2: $A = \begin{bmatrix} 2 & 0 & 1 & -1\\ 0 & 1 & 2 & 0\\ 1 & 2 & 3 & -2 \end{bmatrix}$, $\vec{x} = \begin{bmatrix} 2\\ 3\\ -1\\ 4 \end{bmatrix}$ find $A\vec{x}$.

Example 3: Write $-3\vec{u} + 2\vec{v}$ as a matrix times a vector where $\vec{u} = \begin{bmatrix} 2\\-1\end{bmatrix}$ and
$\vec{v} = \begin{bmatrix} -3\\4\end{bmatrix}$.

Exercise 3: Write $4\vec{u} + (-2)\vec{v} + 3\vec{w}$ as a matrix times a vector where $\vec{u} = \begin{bmatrix} -1\\-3\end{bmatrix}$,
$\vec{v} = \begin{bmatrix} -2\\0\end{bmatrix}$ and $\vec{w} = \begin{bmatrix} 0\\2\end{bmatrix}$.

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

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

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

Definition: If $A$ is an $m \times n$ matrix, with column vectors, $\vec{v_{1}}, \cdots,\vec{v_{n}}$ in $\mathbb{R}^m$, and $\vec{x}$ is a vector in $\mathbb{R}^n$, and $\vec{b}$ is a vector in $\mathbb{R}^m$, then $A\vec{x} = \vec{b}$ is called a
matrix equation.

Theorem: If $A$ is an $m \times n$ matrix, with column vectors, $\vec{v_{1}}, \cdots, \vec{v_{n}}$ in $\mathbb{R}^n$, and $\vec{x}$ is a vector in $\mathbb{R}^n$, and $\vec{b}$ is a vector in $\mathbb{R}^m$, then the matrix equation $A\vec{x} = \vec{b}$ has the same solution set as the vector equation $x_{1}\vec{v_{1}} + \cdots + x_{n}\vec{v_{n}} = \vec{b}$, which, in turn, has the same solution set as the system of linear equations whose augmented matrix is $\begin{bmatrix} \vec{v_{1}} & \cdots & \vec{v_{n}} & \vec{b} \end{bmatrix}$.

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

Example 4: Solve the matrix equation $A\vec{x} = \vec{b}$ where $A = \begin{bmatrix} 2&1\\ 1& -2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 3\\ 4\end{bmatrix}$.

Exercise 4: Solve the matrix equation $A\vec{x} = \vec{b}$ where $A = \begin{bmatrix} 1&1\\ 3& -2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 1\\ -1\end{bmatrix}$.

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 $\vec{b}$ in $\mathbb{R}^m$, the equation $A\vec{x} = \vec{b}$ has a solution

2. Each $\vec{b}$ 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 $\vec{b}$ in $\mathbb{R}^m$ is a linear combination of the columns of $A = \begin{bmatrix} 1&-1&2\\ 2&0&3\\ 0&1&-2 \end{bmatrix}$

Exercise 5: Show each $\vec{b}$ in $\mathbb{R}^m$ is a linear combination of the columns of $A = \begin{bmatrix} 0&-1&2\\ 1&3&0\\ -2&2&-2 \end{bmatrix}$

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 $I_{n}\vec{x} = \vec{x}$ for each n-vector $\vec{x}$ in $\mathbb{R}^n$.

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

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

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

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

c. The equation $A\vec{x} = \vec{b}$ is consistent if the augmented matrix $\begin{bmatrix} A & \vec{b}\end{bmatrix}$ has a pivot position in every row.

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

e. If $A$ is an $m \times n$ matrix and if the equation $A\vec{x} = \vec{b}$ is inconsistent for some $\vec{b}$ 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 $c_{1}\vec{v_{1}} + \cdots + c_{p}\vec{v_{p}}$ cannot all be zero.

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

$A = \begin{bmatrix} 1 & 3 & 3 & 0\\ 0 & 0 & -1 & 1\\ 0 & 1 & 2 & 1 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} -2\\ 1\\ 1 \end{bmatrix}$

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

a. $\begin{bmatrix}3\\2\end{bmatrix}$ is a linear combination of $\begin{bmatrix}1\\0\end{bmatrix}$ and $\begin{bmatrix}0\\1\end{bmatrix}$

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

c. If $A\vec{x} = 0$ where $\vec{x} \neq 0$, then $A = 0$.

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

e. If $A = \begin{bmatrix}\vec{a_{1}}&\vec{a_{2}}&\vec{a_{3}}\end{bmatrix}$ in terms of its columns, and if $\vec{b} = 3\vec{a_{1}} -2\vec{a_{2}}$, then the system $A\vec{x} = \vec{b}$ has a solution.

f. If $A = \begin{bmatrix}\vec{a_{1}}&\vec{a_{2}}&\vec{a_{3}}\end{bmatrix}$ in terms of its columns, and if $A\vec{x} = \vec{b}$ has a solution then
$\vec{b}=s\vec{a_{1}} - t\vec{a_{2}}$ for some $s, t$.

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

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

## 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 $\vec{x}$ in $\mathbb{R}^n$ to a vector $T(\vec{x})$ in $\mathbb{R}^m$.

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

3. $T(\vec{x})$ is called the image of $\vec{x}$. The set {$T(\vec{x})$ for any $\vec{x}$ 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 $T(\vec{x}) = A\vec{x}$. This kind of transformation is called a matrix transformation. $T$ is the matrix transformation induced by A.

Example 1: Let $A = \begin{bmatrix} 1 & -2\\ -2 & 4\\ 3 & -6 \end{bmatrix}$, $\vec{b} = \begin{bmatrix} 3\\ 4\end{bmatrix}$, $\vec{c} = \begin{bmatrix} 3\\ -6\\ 9\end{bmatrix}$, and $\vec{d} = \begin{bmatrix} 2\\ -6\\ 9\end{bmatrix}$.

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

(a) Find $T(\vec{b})$. (b) Find an $\vec{x}$ such that the image under $T$ is $\vec{c}$. (c) Is there more than one $\vec{x}$ such that whose image under $T$ is $\vec{c}$? (d) Is there an $\vec{x}$ such that its image under $T$ is $\vec{d}$?

Exercise 1: Let $A = \begin{bmatrix} -1 & 3\\ 2 & -5\\ 2 & -4 \end{bmatrix}$, $\vec{b} = \begin{bmatrix} 5\\ 6\end{bmatrix}$, $\vec{c} = \begin{bmatrix} -2\\ 5\\ 6\end{bmatrix}$, and $\vec{d} = \begin{bmatrix} 2\\ -6\\ 5\end{bmatrix}$.

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

(a) Find $T(\vec{b})$. (b) Find an $\vec{x}$ such that the image under $T$ is $\vec{c}$. (c) Is there more than one $\vec{x}$ such that whose image under $T$ is $\vec{c}$? (d) Is there an $\vec{x}$ such that its image under $T$ is $\vec{d}$?

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

Definition: 1. Given a matrix $A = \begin{bmatrix} 1 & m\\ 0 & 1\end{bmatrix}$, where $m$ is a constant then the transformation $T$ defined by $T(\vec{x}) = A\vec{x}$ is called a shear transformation. This kind of transformation send the vector $\begin{bmatrix} x_{1}\\x_{2}\end{bmatrix}$ to $\begin{bmatrix} x_{1} + mx_{2}\\x_{2}\end{bmatrix}$, i.e sending the point $(x_{1}, x_{2})$ to the point $(x_{1} + mx_{2}, x_{2})$. $(x_{1}, x_{2})$ is moving horizontally to the right if $m > 0$ and $x_{2} > 0$ or $m 0$ and $x_{2} 0$, otherwise is moving horizontally to the left.

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

Example 2: Find the image of $\vec{u} = \begin{bmatrix} 2\\ -3\end{bmatrix}$ and $\vec{v} = \begin{bmatrix} -2\\ 3\end{bmatrix}$ under the matrix transformation defined by $A = \begin{bmatrix} 1 & 4\\0 & 1\end{bmatrix}$. Then graph the corresponding points on the plane.

Exercise 2: Find the image of $\vec{u} = \begin{bmatrix} -1\\ 2\end{bmatrix}$ and $\vec{v} = \begin{bmatrix} 2\\ -1\end{bmatrix}$ under the matrix transformation $T$, defined by $A = \begin{bmatrix} 1 & -3\\0 & 1\end{bmatrix}$. Describe geometrically what $T$ does to each vector.

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

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

Definition: The transformation

$R_{\theta} : \mathbb{R}^2\rightarrow \mathbb{R}^2$

denotes counterclockwise rotation about the origin through an angle of $\theta$.

Example 4: The transformation

$R_{\pi/2} : \mathbb{R}^2\rightarrow \mathbb{R}^2$

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

$R_{\pi} : \mathbb{R}^2\rightarrow \mathbb{R}^2$

denotes a counterclockwise rotation about the origin through an angle of
$\pi$ radians. Find the matrix of $R_{\pi}$.

Example 5: Let $T(\vec{x}) = A\vec{x}$ where $\begin{bmatrix} 1 & 0 & 5\\ 2 & 3 & 0\end{bmatrix}$. Find a vector $\vec{x}$ such that $T(\vec{x}) = \vec{b} = \begin{bmatrix} 3\\ -1\end{bmatrix}$.

Exercise 5: Let $T(\vec{x}) = A\vec{x}$ where $\begin{bmatrix} 0 & 1 & -3\\ -2 & 2 & 0\end{bmatrix}$. Find a vector $\vec{x}$ such that $T(\vec{x}) = \vec{b} = \begin{bmatrix} -2\\ 1\end{bmatrix}$.

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

e. If $\vec{x_{1}}$ and $\vec{x_{2}}$ are solutions to $A\vec{x} = \vec{b}$, then $\vec{x_{1}}$ – $\vec{x_{2}}$ is a solution to $A\vec{x} = \vec{0}$.

f. Let $A = \begin{bmatrix}\vec{a_{1}}\vec{a_{2}}\vec{a_{3}}\end{bmatrix}$ in terms of its columns. If $\vec{a_{3}} = s\vec{a_{1}} + t\vec{a_{2}}$, then $A\vec{x} = 0$, where $\vec{x} = \begin{bmatrix}s\\t\\-1\end{bmatrix}$.

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

Let $T: \mathbb{R}^2\rightarrow \mathbb{R}^2$ 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 $\vec{b}$ is a linear combination of the columns of $A$, then the system $A\vec{x} = \vec{b}$ is consistent.

b. Let $\vec{x_{1}}$ and $\vec{x_{2}}$ be solutions to the homogeneous system $A\vec{x} = \vec{0}$ then $\vec{x_{1}} + \vec{x_{2}}$ is a solution to $A\vec{x} = \vec{0}$.

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

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

## License

Matrices Copyright © by Kuei-Nuan Lin is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.