Linear Maps and Hom spaces

We suppose that $$V$$ and $$W$$ are vector spaces over the same field $$F$,$ and $\tau$ is a function from $$V$$ to $$W$.$ In this situation, $\tau$ is said to preserve addition if $\tau(u)=s$ and $\tau(v)=t$ implies $\tau(u+v)=s+t$ for all $u,v\in V$ and $$s,t\in W$.$ Similarly, $\tau$ is said to preserve scalar multiplication if $\tau(v)=w$ implies $\tau(av)=aw$ for all $$a\in F$,$ $$v\in V$,$ and $$w\in W$.$ If $\tau$ preserves both operations, then it is said to be linear. More generally, functions that preserve the operations defining an algebraic structure are called homomorphisms.

Definition 2.1.1. Let $$V$$ and $$W$$ be vector spaces over the same field $$F$.$ A function $\tau\colon V\to W$ is called a linear map from $$V$$ to $$W$$ if for all $u,v\in V$ and $a\in F$ the following conditions hold:

$$\tau(u+v)=\tau(u) +\tau(v)$;$
$$\tau(av) = a\tau(v)$.$

The set of all linear maps from $$V$$ to $$W$$ is denoted by $$\operatorname{Hom}(V,W)$.$ A linear map from $$V$$ to $$V$$ is called a linear operator on $$V$.$ The identity operator $\iota_V$ on $$V$$ is a linear operator on $$V$$ such that $\iota_V(v)=v$ for all $$v\in V$.$ The set of all linear operators on $$V$$ is denoted by $$\operatorname{End}(V)$.$ A linear operator on a real vector space is called a real operator.

Proposition 2.1.2. Let $$V$$ and $$W$$ be vector spaces over $$F$,$ and let $\tau\colon V \to W$ be a function.

If $\tau$ is linear, then $$\tau(0_V)=0_W$.$
If $\tau$ is linear, then $\tau(-v)=-\tau(v)$ for all $$v\in V$.$
The function $\tau$ is linear if and only if, for all $u,v\in V$ and $$a\in F$,$ we have
\[\tau(au+v)=a\tau(u)+\tau(v).\]

Proof. (1) We have $$v+0_V=v$$ for all $$v\in V$.$ Since $\tau$ is linear, $$\tau(v) + \tau(0_V) = \tau(v)$.$ Since the zero vector in $$W$$ is unique, $$\tau(0_V)=0_W$.$ (2) Since $$v+(-v)=0_V$,$ we have $\tau(v)+\tau(-v)=0_W$ by (1). Since the inverse of a vector is unique, $$\tau(-v)=-\tau(v)$.$ (3) If $\tau$ is linear, then $$\tau(au+v)=\tau(au) + \tau(v)=a\tau(u) + \tau(v)$.$ We now prove the converse. Setting $$a=1$,$ we obtain $$\tau(u+v)=\tau(u)+\tau(v)$.$ By (1), setting $$v=0_V$$ gives $$\tau(au)=a\tau(u)$.$ Therefore $\tau$ is linear. $\square$

We first look at some examples of real operators.

Example 2.1.3. For any $$\theta\in \mathbb R$,$ we define $\tau_{\theta}\colon \mathbb R^2\to \mathbb R^2$ by

\[\tau_{\theta}(x,y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta).\]

The operator $\tau_{\theta}$ is linear. Indeed, we have

\[\begin{align} \tau_{\theta}(a(x,y)+(z,w))&= \tau_{\theta}(ax+z,ay+w) \\ &= ((ax+z)\cos\theta - (ay+w)\sin\theta, (ax+z)\sin\theta + (ay+w)\cos\theta)\\ &= a(x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta) + (z\cos\theta - w\sin\theta, z\sin\theta + w\cos\theta) \\ &= a\tau_{\theta}(x,y) + \tau_{\theta}(z,w). \end{align}\]

The real operator $\tau_{\theta}$ is called the rotation by $\theta$ in dimension $$2$.$

Example 2.1.4. We define $\tau\colon \mathbb R^2\to \mathbb R^2$ by $$\tau(x,y) = (x,-y)$.$ Then $\tau$ is a real operator, and it is called the reflection about the $$x$-$ axis, the set $$\{(x,y)\in \mathbb R^2 : y=0\}$.$

Example 2.1.5. We define $\tau\colon \mathbb R^2\to \mathbb R^2$ by $$\tau(x,y)=(x,0)$.$ Then $\tau$ is a real operator, and it is called the projection on the $$x$-$ axis.

In real analysis, differentiation and integration provide us with two of the most important examples of linear map.

Example 2.1.6. Let $C^{\infty}$ denote the set of all real-valued functions defined on $\mathbb R$ that have derivatives of every order. Then $C^{\infty}$ is a vector space. Define $D\colon C^{\infty}\to C^{\infty}$ by $$Df=f'$.$ Then, for $$f,g\in C^{\infty}$,$

\[D(af+g) = af'+g' = aDf + Dg.\]

Hence $$D$$ is linear. The linear operator $$D$$ is called a differential operator.

Example 2.1.7. Let $$I$$ be an interval, and define $\tau\colon C^1(I)\to \mathbb R$ by

\[\tau(f) =\int_I f(x)\,dx.\]

Then $\tau$ is a linear map.

Theorem 2.1.8. Let $$V$$ and $$W$$ be vector spaces, let $$B$$ be a basis for $$V$,$ and let $$\tau, \sigma\in \operatorname{Hom}(V,W)$.$ If $\tau(b)=\sigma(b)$ for all $$b\in B$,$ then $$\tau=\sigma$.$

Theorem 2.1.9. Let $$V$$ and $$W$$ be vector spaces, and let $$\tau\in \operatorname{Hom}(V,W)$.$

If $$U$$ is a subspace of $$W$,$ then $\tau^{-1}(U)$ is a subspace of $$V$.$
If $$U$$ is a subspace of $$V$,$ then $\tau(U)$ is a subspace of $$W$.$

Proof. (1) Since $\tau(0_V)=0_W$ and $$0_W\in U$,$ $$0_V\in \tau^{-1}(U)$.$ Let $$u,v\in \tau^{-1}(U)$.$ Then we have $\tau(u)\in U$ and $$\tau(v)\in U$.$ Since $$U$$ is a subspace, $$\tau(u)+\tau(v)\in U$.$ Since linearity gives $$\tau(u+v)=\tau(u)+\tau(v)$,$ $$u+v\in \tau^{-1}(U)$.$ Similarly, closedness of $$U$$ under scalar multiplication and linearity of $\tau$ gives $$av\in \tau^{-1}(U)$,$ where $$a\in F$.$ By Theorem 1.3.2, $\tau^{-1}(U)$ is a subspace of $$V$.$ (2) Since $$0_V\in U$,$ $0_W\in \tau(U)$ by Proposition 2.1.2(1). Let $u,v\in \tau(U)$ and let $s\in \tau^{-1}(\{u\})$ and $$t\in \tau^{-1}(\{v\})$.$ Since $$s+t\in U$,$ linearity gives $$u+v=\tau(s+t)\in \tau(U)$.$ Similarly, since $$as\in U$,$ we have $$au=\tau(as)\in \tau(U)$,$ where $$a\in F$.$ By Theorem 1.3.2, $\tau(U)$ is a subspace of $$W$.$ $\square$