Symmetric Parallelograms Under Linear Transfer Marticies

Question

I am trying to show that a parallelogram which is symmetric about the origin stays symmetric about the origin under the action of a linear transfer matrix.

It is a fairly trivial case to draw a picture and convince yourself of this but a mathematical proof eludes me. I am thinking something with the Parallelogram Law of Vector Addition, but have made no progress with it.

Thank

Answer 1

Any set $X$ is symmetric around the origin if $x\in X$ implies $-x\in X$. So you should demonstrate that if this property holds prior to your transformation, then it holds afterwards. And if you have a different precise definition of a parallelogram which is symmetric around the origin, you might also want to show that the two definitions agree.