Question: Hey Exchange community,

We have finite-dimensional vector spaces $V$ and $W$ with ordered bases $B$ and $A$ respectively.

Is it true/false that $L(V,W)=L(W,V)$?

Just wondering if someone could articulate what the difference is between the two notations. Having someone write it down as if they were verbally explaining it to someone would really help.

Thanks!

Answer: 
The symbols $\mathcal{L}(V,W)$ indicates the linear transformations from vector space $V$ into vector space $W$, and the analogous statement is made for $\mathcal{L}(W,V)$. If $V$ and $W$ are not the same space (e.g. isomorphic) then you don't in general have the same maps going from $V$ to $W$ as you do from $W$ to $V$. For instance, if $V$ were $2$-dimensional and $W$ were $3$-dimensional then the linear maps in $\mathcal{L}(V,W)$ could be represented by $3\times 2$ matrices, but the maps in $\mathcal{L}(W,V)$ would be $2\times 3$ matrices.
$\mathcal{L}(U,W)=\{T\colon U\to W/ T \mbox{ is a linear transformation }\}$
$\mathcal{L}(W,U)=\{T\colon W\to U/ T \mbox{ is a linear transformation }\}$

There are different because $Im(\mathcal{L}(U,V))\subset U$ and $Im((\mathcal{L}(V,U))\subset V$ and $ U, V$ nor always will be subsets of the same space. 

For example $\mathcal{L}(U,\mathbb{K})$ obviusly not equal to $\mathcal{L}(\mathbb{K},U)$