Geometric interpretation of non-square matrices I realize that a $n\times{n}$ matrix can be interpreted as linear transformation of a vector in n-dimensional coordinate system. 
But I am not able to interpret any $m\times{n}$ matrix same way since $m\times{n}$ doesn't mention all the coordinates. How  can this type of transformation be visualized?
 A: Matrices are a concise way of writing linear transformations. Matrix multiplication of a matrix $M\in\mathbb{R}^{n\times n}$ and a vector $x\in\mathbb{R}^n$ results in a new vector $x'\in\mathbb{R}^n$ that lies in the same space as the original vector (i.e. $Mx=x'$). 
Now consider the multiplication of a matrix $M\in\mathbb{R}^{m\times n}$ and a vector $x\in\mathbb{R}^n$; the resulting vector $x'$ no longer lies in $\mathbb{R}^n,$ but in $\mathbb{R}^m$ (i.e. $x$ has been mapped from $\mathbb{R}^n$ to $\mathbb{R}^m$).
For intuition on what this means, consider the matrix multiplication of a matrix $M\in\mathbb{R}^{1\times n}$ and a vector $x\in\mathbb{R}^n.$ The results $x'$ lines in $\mathbb{R}$--the vector has been mapped from an $n$-dimensional vector to a scalar value! 
Your concern about the result not having the same dimensions as the original is not unfounded--indeed, we have lost potentially valuable information about the original vector by representing it in a lower dimensional space.
