Linear transformation notation question

Question

Well the question is pretty basic, but I am learning math on my own. And I cannot understand the notation of linear transformation. I understand what linear transformation is, its properties and what not. I could not find the answer elsewhere.

What does $L_A \colon \mathbb{R}^n \to \mathbb{R}^m$ mean exactly? I have problems reading this part: $\mathbb{R}^n \to \mathbb{R}^m$

I think it would be more clear if it was written like: $L_A \colon V \to W$, where $V$ and $W$ are vectors.

Are there other ways to express this?

Answer 1

It means that $L_A$ is a mapping (or function) from the space $\mathbb{R}^n$ to the space $\mathbb{R}^m$. In other words, for any given vector $v$ in $\mathbb{R}^n$, the mapping $L_A$ gives you a vector $L_A(v)$ in $\mathbb{R}^m$ as output.

This is just the standard way of writing functions.

You write

• The name of the function (in this case $L_A$)
• A colon
• The space that the function maps from (called its domain)
• An arrow
• The space that the function maps into (called its range or codomain)

Just some jargon you have to get used to, I'm afraid.

Answer 2

The vector spaces $\mathbb{R}^n$ and $\mathbb{R}^m$ are the domain and codomain of the function respectively. This means that $L_A$ takes as inputs, elements from the vector space $\mathbb{R}^n$, and outputs elements of the vectors space $\mathbb{R}^m$.