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?

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
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

score 1 · Accepted Answer · answered Mar 22 '15 at 13:08

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.

answered Mar 22 '15 at 13:08

bubba

43,483
3
61
122

Thanks. To clarify. If for example m=n, vector stays in the same subspace? Does superscript indicates dimensionality? Like R², 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

score 0 · Answer 2 · answered Mar 22 '15 at 13:06

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 $\mathbb{R}^m$.

Linear transformation notation question

2 Answers2