Linear transformation, volume, problem
True or false :
Let be a linear transformation defined by , , .
If is the cube formed by the vertices , , , , then .
My attempt : By intuition its false because seems more like an area, but it could be a volume.
Ok, so I have formed in order to find out that . Then I found the vertices passed by the transformation to be , , and . Now it remains to calculate the volume of the solid formed by these vertices. I'm not really good when it comes to draw 3 dimensional solids. Especially this "prism" looking volume. I've no idea how to find out its volume.
Oh.... let me see, I remember I've read in a calculus III book that I must calculate a jacobian matrix and check out the determinant, or something like that. If it's 1 then the volume of the cube is the same as the volume of this prism. And if it's 2 then the volume of the prism is half or twice the volume of the cube, etc. I don't know which jacobian to find though.