# Linear transformation, volume, problem

• Jul 4th 2009, 04:30 PM
arbolis
Linear transformation, volume, problem
True or false :
Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation defined by $T(1,0,0)=(1,1,0)$, $T(0,1,0)=(-2,1,1)$, $T(0,0,1)=(0,3,-2)$.
If $C$ is the cube formed by the vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, then $T(C)=3^2$.
My attempt : By intuition its false because $3^2$ seems more like an area, but it could be a volume.
Ok, so I have formed $[T]_c^{c}$ in order to find out that $T(x,y,z)=(x-2y,x+y+3z,y-2z)$. Then I found the vertices passed by the transformation to be $(0,0,0)$, $(1,1,0)$, $(-2,1,1)$ and $(0,3,-2)$. 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.
• Jul 4th 2009, 06:39 PM
alunw
Write the transformation as a matrix. That's very easy since you have been told what it does to the basis vectors. It is
1 -2 0
1 1 3
0 1 -2
(or the transpose if you want to write vectors using rows, so that you calculate the action by doing v*M rather than M*v)
The determinant of the matrix is the volume of your transformed cube.
• Jul 4th 2009, 08:29 PM
malaygoel
Quote:

Originally Posted by arbolis
$T(C)=3^2$.

Can you explain how?
$T(C)=3^2$
• Jul 5th 2009, 07:34 AM
HallsofIvy
You can do it directly. The three edges of C from (0,0,0) are given by the vectors <1, 0, 0>, <0, 1, 0>, and <0, 0, 1>. Applying T to each of those, we get, as we are told, <1, 1, 0>, <-2, 1, 1>, and <0, 3, -2>. Now use the fact that the volume of such a "parallelopiped" is given by the absolute value of the determinant
$\left|\begin{array}{ccc}1 & 1 & 0 \\ -2 & 1 & 1 \\ 0 & 3 & -2\end{array}\right|$ $= \left|\begin{array}{cc}1 & 1 \\ 3 & -2\end{array}\right|- \left|\begin{array}{cc}-2 & 0 \\ 0 & -2\end{array}\right|$= (-2-3)- 4= -9
so the absolute value is $9= 3^2$. The fact that a number happens to be a perfect square doesn't mean it must be an area!