Before entering my problem, let me review some related results: 


Suppose $\mathcal{S}$ is a convex hull of finite points: $\mathcal{S}=\operatorname{conv}(x\_1,x\_2,\ldots,x\_m)$, then 


1. By "<https://math.stackexchange.com/questions/282036/convex-hull-of-extreme-points>" , we know $\mathcal{S}$ is the convex hull of its extreme points, which is the subset of $\{x\_1,\ldots, x\_m\}$.
2. By "<https://math.stackexchange.com/questions/1591561/preservation-of-extreme-points-under-linear-transformation>", if the transformation is linear, let $A$ = {the extreme points of the image of a compact convex set}, then the **preimage** of $A$ must be the extreme points of that compact convex set. But the **inverse** in not true. The inverse is true only under such transformation 
is injective and surjective.


My question is: 


Suppose 


1. We have two sets $A,B$ with $L: A\rightarrow B$, which is **linear**.
2. $A,B$ are both convex hull of finite extreme points, compact and both sets have at least one extreme point.
3. Suppose $L$ maps **extreme points** in $A$ **one-to-one and onto** to **extreme points** in $B$.


Question: Is $L$ a one-to-one and onto linear map for $A\rightarrow B$? 


If not, could you please provide an counter-example?




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Moreover, if $A,B$ are convex hull of **infinite extreme points**, will I get the same answer?


This part comes from an example: 


The extreme points of $A$ can be $SO(2)$ 
, i.e. a set of $2×2$ rotation matrices with $0≤θ≤2π$, to make an one-to-one map. The extreme points of $B$ is a set of rank one, $2×2$ positive semidefinite matrices with trace one. So $B$ 
 can be built by $vv^T$ where $v = [v\_1 v\_2]^T$
 with $v\_1^2 +v\_2^2 =1$. 



Linear map of finite or infinite extreme points. Discuss injectivity and surjectivity
So, you should try to map the vertices of a tetrahedron linearly onto the vertices of a square. For example by a projection $\mathbb R^3 \to \mathbb R^2$.


