In linear algebra, we study vectors in some space Rn\mathcal{R}^n. Linear transformation is defined on Rn\mathcal{R}^n. For v,u∈Rnv, u \in \mathcal{R}^n, and c∈Rc \in \mathcal{R}, a linear transformation T:Rn→RnT : \mathcal{R}^n \rightarrow \mathcal{R}^n satisfies

  1. T(v+u)=T(v)+T(u)T(v + u) = T(v) + T(u)
  2. T(cv)=cT(v)T(cv) = c T(v)

However, the "linear thing" is not limited to vectors. As an illustration, note that a vector

v=[v1v2...vn] v = \begin{bmatrix} v_1 \\ v_2 \\ . \\ . \\ . \\ v_n \end{bmatrix}

can be viewed as the image of some function f:{1,2,...,n}→Rf : \{1, 2, ..., n \} \rightarrow \mathcal{R}, such that f(i)=vif(i) = v_i. It is natural to study the functions {I→R}\{\mathbb{I} \rightarrow \mathcal{R} \}, where I⊂R\mathbb{I} \subset \mathcal{R} is some open interval. For any x∈Ix \in \mathbb{I}, f(x)f(x) gives the value of xx. The only difference with vectors is that, it not possible to enumerate all value of x∈Ix \in \mathbb{I}. Instead, we specify a rule of obtaining the value at xx.

For any two functions f1,f2:I→Rf_1, f_2 : \mathbb{I} \rightarrow \mathcal{R}, we can add them together and obtain a new function f1+f2:I→Rf_1 + f_2 : \mathbb{I} \rightarrow \mathcal{R}. Addition is closed in the space {I→R}\{\mathbb{I} \rightarrow \mathcal{R} \}. Then for x∈Ix \in \mathbb{I},

(f1+f2)(x)=f1(x)+f2(x) (f_1 + f_2) (x) = f_1(x) + f_2(x)

which is similar to that adding up coordinates of two vector. We can also define multiplication: for c∈Rnc \in \mathcal{R}^n, f:I→Rf : \mathbb{I} \rightarrow \mathcal{R}, cfcf is a function in {I→R}\{\mathbb{I} \rightarrow \mathcal{R} \} such that

(cf)(x)=cf(x) (cf)(x) = c f(x)

Multiplication is closed in the space {I→R}\{\mathbb{I} \rightarrow \mathcal{R} \}.

Example. The derivative operation T:{f:I→R∧differentiable}→{I→R}T: \{f: \mathbb{I} \rightarrow \mathcal{R} \wedge \text{differentiable} \} \rightarrow \{\mathbb{I} \rightarrow \mathcal{R} \} is linear transformation. It is easy to verify for differentiable functions f1,f2∈I→Rf_1, f_2 \in \mathbb{I} \rightarrow \mathcal{R}, and c∈Rc \in \mathcal{R}, we have

  1. (f1+f2)′=f1′+f2′(f_1 + f_2)' = f_1' + f_2'
  2. (cf1)′=cf1′(c f_1)' = c f_1'

What is more, we can extend some other concepts in linear algebra. In linear algebra, "eigenvector" is a vector whose direction does not change under a linear transformation. For derivative operation, the function exe^x is un-changed after the transformation:

(ex)′=ex (e^x)' = e^x