TITLE: Can I have a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
QUESTION [0 upvotes]: Is it possible to have a linear map $T: \mathbb{R}^m \to \mathbb{R}^n$ where $m\ne n$ for an operator?
I'm curious because on an assignment question before it wanted me to check if an operator was linear  (the projection operator) but before it did, it specified $T$ as above except that it was $\mathbb{R}^m \to \mathbb{R}^m$.

REPLY [0 votes]: This is - as already pointed out in the comment - totally feasible. The linear maps between $\mathbb{R}^n$ and $\mathbb{R}^m$ are exactly the matrices $M$ of the form $M\in \mathbb{R}^{m\times n}$. There is a one-to-one relation between matrices and linear maps when it comes to finite dimensional vector spaces.
So basically any matrix $M$ with $m$ rows and $n$ columns will represent such a linear map.
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