Interpretation of Linear Algebra and superposition

Question

I have been told and have seen that knowing Linear Algebra is foundational in pursuing advanced mathematics. After dealing with it enough I have gotten "used to it" (I still need a lot more practice!) What I would like to know is the interpretation of Linear Algebra so that I may have more mathematical intuition when dealing with it.

Specifically, a transformation $L$ is linear if it satisfies the properties:

$aL(x) = L(ax)$ and $L(x+y)= L(x) + L(y)$ where $a$ is a scalar and $x,y$ are vectors.

1. How can I think about this intuitively? (Why does the term superposition accurately fit this?)
2. How else can one interpret results in Linear Algebra?

Answer 1

Let $V$ and $W$ be vector spaces, and $L:V\to W$ a linear map. By either of the properties, we get $L(0)=0$.

Geometrically, a linear map assigns a vector in $W$ to each vector of $V$, so that

1. The origin goes to the origin. ($L(0)=0\,$.)
2. Lines go to lines. (Because $L(x_0+tv)=L(x_0)+t\,L(v)$ for any $t\in\Bbb R$.)
3. Triangles go to triangles (any two vector $x,y$ determines a triangle together with $x+y$, so this corresponds to $L(x+y)=L(x)+L(y)$).
4. It is invariant under zooming from the origin. ($L(ax)=a\,L(x)\,$.)