Let $V, W$ be two vector spaces. A map $L:V\longrightarrow W$ is called a linear map if it satisfies the following properties: for any elements $u,v\in V$ and any scalar $c$,

LM 1. $L(u+v)=L(u)+L(v)$.

LM 2. $L(cu)=cL(u)$.

That is, linear maps are maps that preserve addition and scalar multiplication.

Proposition. A map $L: V\longrightarrow W$ is linear if and only if for any elements $u,v\in V$ and scalars $a,b$,
$$L(au+bv)=aL(u)+bL(v).$$

Proof. It is straightforward and left as an exercise.

Example. Let $A$ be an $m\times n$ matrix. Define
$$L_A:\mathbb{R}^n\longrightarrow\mathbb{R}^m$$
by
$$L_A(X)=A\cdot X.$$ Then $L_A$ is linear.

Example. Let $A=(a_1,\cdots,a_n)$ be a fixed vector in $\mathbb{R}^n$. Define $L_A:\mathbb{R}^n\longrightarrow\mathbb{R}$ by
$$L_A(X)=A\cdot X.$$
Then $L_A$ is a linear map. The dot product $A\cdot X$ can be viewed as a matrix multiplication if we view $A$ as a row vector and $X$ as a column vector. So this example is a spacial case of the previous example.

Example. Let $\mathcal{F}$ be the set of all smooth functions. Then the derivative $D:\mathcal{F}\longrightarrow\mathcal{F}$ is a linear map.

Example. Define $\wp: \mathbb{R}^3\longrightarrow\mathbb{R}^2$ by $\wp(x,y,z)=(x,y)$, i.e. $\wp$ is a projection. It is a linear map.

Proposition. Let $L: V\longrightarrow W$ be a linear map. Then $L(O)=O$.

Proof. Let $v\in V$. Then
$$L(O)=L(v-v)=L(v-v)=L(v)-L(v)=O.$$

Example. Let $L:\mathbb{R}^2\longrightarrow\mathbb{R}^2$ be a linear map. Suppose that
$$L(1,1)=(1,4)\ \rm{and}\ L(2,-1)=(-2,3).$$
Find $L(3,-1)$.

Solution. $(3,-1)$ is written as a linear combination of $(1,1)$ and $(2,-1)$ as
$$(3,-1)=\frac{1}{3}(1,1)+\frac{4}{3}(-2,3).$$
Hence,
$$L(3,1)=\frac{1}{3}L(1,1)+\frac{4}{3}L(-2,3)=\frac{1}{3}(1,4)+\frac{4}{3}(-2,3)=\left(-\frac{7}{3},\frac{16}{3}\right).$$

The coordinates of a linear map

Consider a map $F: V\longrightarrow\mathbb{R}^n$. For any $v\in V$, $F(v)\in\mathbb{R}^n$ so $F(v)$ may be written as
$$F(v)=(F_1(v),F_2(v),\cdots,F_n(v))$$
where each $F_i$ is a function $F_i:V\longrightarrow\mathbb{R}$ called the $i$-th coordinate function.

Proposition. A map $F_i: V\longrightarrow\mathbb{R}^n$ is linear if and only if each coordinate function $F_i$ is linear.

Proof. Straightforward. Left as an exercise.

Example. Let $F:\mathbb{R}^2\longrightarrow\mathbb{R}^3$ be the map
$$F(x,y)=(2x-y,3x+4y,x-5y).$$
Then
$$F_1(x,y)=2x-y,\ F_2(x,y)=3x+4y,\\ F_3(x,y)=x-5y.$$
These coordinate functions can be written as
$$F_1(x,y)=\begin{pmatrix}
2 & -1
\end{pmatrix}\begin{pmatrix}
x\\y
\end{pmatrix},\ F_2(x,y)=\begin{pmatrix}
3 & 4
\end{pmatrix}\begin{pmatrix}
x\\y
\end{pmatrix},\ F_3(x,y)=\begin{pmatrix}
1 & -5
\end{pmatrix}\begin{pmatrix}
x\\y
\end{pmatrix}.$$
Hence, each $F_i$ is linear, $i=1,2,3$ and therefore $F$ is linear by the Proposition. In fact, $F$ may be written as $L_A:\mathbb{R}^2\longrightarrow\mathbb{R}^3$ where
$$A=\begin{pmatrix}
2 & -1\\
3 & 4\\
1 & -5
\end{pmatrix}.$$