Introduction

Section 3.1 Introduction

Consider the Cartesian plane \(\R^2=\left\{\begin{bmatrix}x_1\\x_2 \end{bmatrix} :x_1,x_2\in\R\right\}\text{.}\) The reflection about \(x\)-axis in \(\R^2\) is given by \(R_f\colon \R^2\to\R^2\)

\begin{equation*} R_f\left(\begin{bmatrix}x_1\\x_2 \end{bmatrix} \right)=\begin{bmatrix}x_1\\-x_2 \end{bmatrix}\text{.} \end{equation*}

Look at the Figure 3.1.1.

Figure 3.1.1. Reflection of a point about \(x\)-axis.
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Note that we can write

\begin{equation*} R_f\left(\begin{bmatrix}x_1\\x_2 \end{bmatrix} \right)= \begin{bmatrix}x_1\\-x_2 \end{bmatrix} = \begin{bmatrix}1 \amp 0\\0 \amp -1 \end{bmatrix} \begin{bmatrix}x_1\\x_2 \end{bmatrix}\text{.} \end{equation*}

Thus the map \(R_f\) is obtained as a matrix multiplication by \(A=\begin{bmatrix}1 \amp 0\\0 \amp -1 \end{bmatrix}\text{.}\)

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Similarly consider a map \(R_\theta\) as a rotation of a vector by an angle \(\theta\) anti-clock wise. Any point \(x=\begin{bmatrix}x_1\\x_2 \end{bmatrix} \in\R^2\) can be written as \(\begin{bmatrix}r\cos\alpha\\r\sin\alpha \end{bmatrix}\) in polar coordinates. Here \(r=\sqrt{x_1^2+x_2^2}\) and \(\alpha\) is the angle that vector \(x\) makes with positive \(x\)-axis. Then

\begin{equation*} R_\theta\left(\begin{bmatrix}x_1\\x_2 \end{bmatrix} \right)= \begin{bmatrix}r\cos(\alpha+\theta)\\r\sin(\alpha+\theta) \end{bmatrix}\text{.} \end{equation*}

Look at the Figure 3.1.1.

Figure 3.1.2. Rotation of a point by \(\theta\text{.}\)
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After Simplification, we have

\begin{equation*} \begin{bmatrix}r\cos(\alpha+\theta)\\r\sin(\alpha+\theta) \end{bmatrix} = \begin{bmatrix}r(\cos \theta \cos\alpha -\sin\theta \sin \alpha) \\ r(\sin\theta\cos\alpha+\cos\theta\sin\alpha) \end{bmatrix} = \begin{bmatrix}\cos \theta \amp -\sin \theta\\ \cos \theta \amp \sin \theta \end{bmatrix} \begin{bmatrix}r\cos\alpha\\r\sin\alpha \end{bmatrix} \end{equation*}

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Thus

\begin{equation*} R_\theta\left(\begin{bmatrix}x_1\\x_2 \end{bmatrix} \right)= \begin{bmatrix}\cos \theta \amp -\sin \theta\\ \cos \theta \amp \sin \theta \end{bmatrix} \begin{bmatrix}x_1\\x_2 \end{bmatrix}\text{.} \end{equation*}

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Thus the rotation in the plane can also be given by the matrix multiplication.

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Suppose \(A\) is a \(m\times n\) real matrix. Then for a vector \(x=\begin{bmatrix}x_1\\\vdots \\x_n \end{bmatrix} \in\R^n\text{,}\) \(Ax\in \R^m\text{.}\) Thus \(A\) can be thought of as a map that takes vector in \(\R^n\) to a vector in \(\R^m\text{.}\) Let us denote this map as \(T_A\text{.}\) Thus

\begin{equation*} T_A\colon R^n\to \R^m, \text{ defined by } T_A(x)=Ax\text{.} \end{equation*}

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\(T_A\) is called the matrix transformation induced by the matrix \(A\text{.}\) Note that the matrix transformation \(T_A\) has the following properties:

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\(T_A(x+y)=A(x+y)=Ax+Ay=T_A(x)+T_A(y)\) for all \(x,y\in \R^n\text{.}\)
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\(T_A(\alpha x)=A(\alpha x)=\alpha Ax=\alpha T_A(x)\) for all \(\alpha\in \R\) and \(x\in \R^n\text{.}\)
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In particular, \(T_A\) preserves vector addition and scalar multiplication. Any such map is called a linear transformation. We have the following definition.

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Definition 3.1.3.

A map \(T\colon \R^n \to \R^m\) is called a linear transformation (or linear map) if it satisfies the following properties:

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for all \(x,y\in \R^n\text{,}\) \(T(x+y)=T(x)+T(y)\text{.}\)
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for all \(\alpha\in \R\) and \(x\in \R^n\text{,}\) \(T(\alpha x)=\alpha T(x)\text{.}\)
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The following are very simple and trivial properties of linear transformation follows directly from the definition.

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If \(T\colon \R^n\to \R^m\) is a linear map, then \(T(0)=0\text{.}\) That is, \(T\) takes the zero vector in \(\R^n\) to the zero vector in \(\R^m\text{.}\)
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If \(T\colon \R^n\to \R^m\) is a linear map, then \(T(-v)=-T(v)\) for all \(v\in \R^n\text{.}\)
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Example 3.1.4.

Zero map \(T\colon \R^n \to \R^m\) defined by \(T(x)=0\) for all \(x\) is a linear map.
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The identity map \(T\colon \R^n \to \R^n\) given by \(T(x)=x\) for all \(x\in \R^n\) is a linear map.
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The reflection and rotation defined above are linear maps from \(\R^2\) to \(\R^2\text{.}\)
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Any matrix transformation given by \(T_A(x)=Ax\) is a linear map.
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Fix a vector \(a\) in \(\R^n\) and define \(T_a\colon \R^n\to \R^n\) as \(T_a(x)=x+a\text{.}\) Is \(T_a\) a linear map? When is this linear? (The map \(T_a\) is called translation by \(a\).)
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Let \(a,b\in \R\) be two real numbers. Define \(T\colon \R^2\to \R\) as \(T(x,y)=ax+by\text{.}\) It is easy to check that \(T\) is a linear map? Can you generalize this?
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Let \(a_1,\ldots, a_n\) be \(n\) real numbers. Define \(T\colon \R^n\to \R\) by

\begin{equation*} T(x_1,\cdots,x_n)=a_1x_1+\cdots+a_nx_n\text{.} \end{equation*}

It is easy to check that \(T\) is a linear map.

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Example 3.1.5. Linear maps from \(\R\to \R\).

Let us find all linear maps from \(\R\) to \(\R\text{.}\) Suppose \(f\colon \R\to \R\) is a linear map. We know that \(\{1\}\) is basis of \(\R\text{.}\) Then any vector \(x=x\cdot 1\text{.}\) Therefore, \(f(x)=f(x\cdot 1)=xf(1)\text{.}\) Thus if we define \(f(1)=a\in \R\) \(f(x)=ax\text{.}\) If \(f\) is linear map from \(\R\to \R\text{,}\) then there exist a real number \(a\in \R\) such that \(f(x)=ax\text{.}\) Note that in this case \(a=f(1)\text{.}\)

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