# Linear transformation notation 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?

• Welcome to MSE! As a side note, you may find it helpful to peruse our basic MathJax tutorial, where you can learn how to (easily!) typeset math on this forum. – A.P. Mar 22 '15 at 13:07
• By the way: are you familiar with the concept of function? This would help us tailor an answer to your needs. – A.P. Mar 22 '15 at 13:11
• yes I am familiar with the concept. – YKY Mar 22 '15 at 13:33
• great. Thanks for pointing me to the tutorial. I think I will have to come back here soon =) – YKY Mar 22 '15 at 13:52

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.

• Thanks. To clarify. If for example m=n, vector stays in the same subspace? Does superscript indicates dimensionality? Like R<sup>2</sup>, means two dimensions? – YKY Mar 22 '15 at 13:30
• Yes, if $m=n$ then $L_A$ is a mapping from some space to itself. In other words, the "output" vectors have the same dimension as the "input" ones. And, "yes" again ... $\mathbb{R}^n$ is the vector space of dimension $n$. So $\mathbb{R}^2$ is two-dimensional space, which you can think of as a plane. – bubba Mar 22 '15 at 13:36
• To get an R with a superscript "2", you type R, then ^, then 2, and you enclose those three characters between dollar signs. – bubba Mar 22 '15 at 13:38

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 $\mathbb{R}^m$.