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Mathematics LibreTexts

6.1 Definition and elementary properties

Throughout this chapter, \(V \) and \(W \) denote vector spaces over \(\mathbb{F} \). We are going to study functions from \(V \) into \(W \) that have the special properties given in the following definition.

Definition 6.1.1.  A function \(T:V\to W \) is called linear if

\[T(u+v) = T(u) + T(v), ~~ \rm{~for ~all~} u,v\in V , \tag{6.1.1}\]
   

\[T(av) = aT(v), ~~ \rm{~for~ all~} a\in \mathbb{F} \rm{~and~} v\in V . \tag{6.1.2}\]

The set of all linear maps from \(V \) to \(W \) is denoted by \(\mathcal{L}(V,W) \). We also write \(Tv \) for \(T(v) \). Moreover, if \(V = W \), then we write \(\mathcal{L}(V,V) = \mathcal{L}(V) \) and call \(T\) in \(\mathcal{L}(V) \) a linear operator on \(V \).

Example 6.1.2.  

1. The zero map \(0:V\to W \) mapping every element \(v\in V \) to \(0\in W \) is linear.

2. The identity map \(I:V\to V \) defined as \(Iv=v \) is linear.

3. Let \(T:\mathbb{F}[z] \to \mathbb{F}[z] \) be the differentiation map defined as \(Tp(z)=p'(z) \).

Then, for two polynomials \(p(z),q(z)\in\mathbb{F}[z] \), we have

\[ T(p(z)+q(z)) = (p(z)+q(z))' =p'(z)+q'(z)=T(p(z))+T(q(z)). \]

Similarly, for a polynomial \(p(z)\in \mathbb{F}[z] \) and a scalar \(a\in \mathbb{F} \), we have

\[ T(ap(z))=(ap(z))'=ap'(z)=aT(p(z)). \]

Hence \(T \) is linear.
4. Let \(T:\mathbb{R}^2\to \mathbb{R}^2 \) be the map given by \(T(x,y)=(x-2y,3x+y) \). Then, for \((x,y),(x',y')\in \mathbb{R}^2 \), we have
\begin{equation*}
\begin{split}
    T((x,y)+(x',y')) &= T(x+x',y+y') = (x+x'-2(y+y'),3(x+x')+y+y')\\
                        &= (x-2y,3x+y) + (x'-2y',3x'+y') = T(x,y) + T(x',y').
\end{split}                       
\end{equation*}

Similarly, for \((x,y)\) in \(\mathbb{R}^2 \) and \(a\) in \(\mathbb{F}\), we have

\[    T(a(x,y)) = T(ax,ay) = (ax-2ay,3ax+ay) = a(x-2y,3x+y) = aT(x,y). \]

Hence \(T\)  is linear. More generally, any map \(T: \mathbb{F}^n \to \mathbb{F}^m \) defined by

\[    T(x_1,\ldots,x_n) = (a_{11}x_1+\cdots +a_{1n} x_n, \ldots,a_{m1}x_1+\cdots+a_{mn}x_n) \]

with \(a_{ij}\in\mathbb{F} \) is linear.

5. Not all functions are linear! For example, the exponential function \(f(x)=e^x \) is not linear since \(e^{2x} \neq 2 e^x \) in general. Also, the function \(f:\mathbb{F} \to \mathbb{F} \) given by \(f(x)=x-1 \) is not linear since \(f(x+y)=(x+y)-1 \neq (x-1)+(y-1)=f(x)+f(y) \).

       An important result is that linear maps are already completely determined if their values on basis vectors are specified.

Theorem 6.1.3.  Let \((v_1,\ldots,v_n) \) be a basis of \(V \) and \((w_1,\ldots,w_n) \) be an arbitrary list of vectors in \(W \). Then there exists a unique linear map

\[    T:V\to W \quad \text{such that \(T(v_i)=w_i, \, \forall \, i = 1, 2, \ldots, n \).} \]

Proof.  First we verify that there is at most one linear map \(T \) with \(T(v_i)=w_i \). Take any \(v\in V \). Since \((v_1,\ldots,v_n) \) is a basis of \(V \) there are unique scalars \(a_1,\ldots,a_n\) in \(\mathbb{F} \) such that \(v = a_1 v_1 + \cdots + a_n v_n \). By linearity, we have 

\[ \begin{equation} \label{eq:T}
    T(v)=T(a_1 v_1 + \cdots + a_n v_n) = a_1T(v_1)+\cdots+a_nT(v_n)
    =a_1 w_1 + \cdots + a_n w_n, \tag{6.1.3}
\end{equation} \]

and hence \(T(v) \) is completely determined. To show existence, use Equation (6.1.3) to define \(T \). It remains to show that this \(T \) is linear and that \(T(v_i)=w_i \). These two conditions are not hard to show and are left to the reader.

       

        The set of linear maps \(\mathcal{L}(V,W) \) is itself a vector space. For \(S,T\in \mathcal{L}(V,W) \) addition is defined as

\[ \begin{equation*}
    (S+T)v = Sv + Tv, \quad \text{for all \(v\in V \).}
\end{equation*} \]
For \(a\in \mathbb{F} \) and \(T\in \mathcal{L}(V,W) \), scalar multiplication is defined as
\[ \begin{equation*}
    (aT)(v) = a(Tv), \quad \text{for all \(v\in V \).}
\end{equation*} \]

You should verify that \(S+T \) and \(aT \) are indeed linear maps and that all properties of a vector space are satisfied.

        In addition to the operations of vector addition and scalar multiplication, we can also define the composition of linear maps. Let \(V,U,W \) be vector spaces over \(\mathbb{F} \). Then, for \(S\in\mathcal{L}(U,V) \) and \(T\in \mathcal{L}(V,W) \), we define \( T\circ S\in\mathcal{L}(U,W)\)  by

\begin{equation*}
    (T\circ S)(u) = T(S(u)), \quad \text{for all \(u\in U \).}
\end{equation*}

The map \(T\circ S \) is often also called the product of \(T \) and \(S \) denoted by \(TS \). It has the following properties:

1. Associativity: \((T_1 T_2)T_3=T_1(T_2 T_3) \), for all \(T_1\in \mathcal{L}(V_1,V_0) \), \(T_2 \in \mathcal{L}(V_2,V_1) \) and \(T_3\in\mathcal{L}(V_3,V_2) \).

2. Identity: \(TI=IT=T \), where \(T\in \mathcal{L}(V,W) \) and where \(I \) in \(TI \) is the identity map in \(\mathcal{L}(V,V) \) whereas the \(I \) in \(IT \) is the identity map in \(\mathcal{L}(W,W) \).

3.Distributivity: \((T_1+T_2)S=T_1S+T_2S \) and \(T(S_1+S_2)=TS_1+TS_2 \), where \(S,S_1,S_2\in\mathcal{L}(U,V) \) and \(T,T_1,T_2\in\mathcal{L}(V,W) \).
 

       Note that the product of linear maps is not always commutative. For example, if we take \(T\in\mathcal{L}(\mathbb{F}[z],\mathbb{F}[z]) \) to be the differentiation map \(Tp(z)=p'(z) \) and \(S\in\mathcal{L}(\mathbb{F}[z],\mathbb{F}[z])\) to be the map \(Sp(z)=z^2p(z) \), then

\[ \begin{equation*}
    (ST)p(z)=z^2 p'(z) \quad \text{but} \quad (TS)p(z) = z^2 p'(z)+2zp(z).
\end{equation*}\]

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