Linear transformation of the line

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October 7th 2011, 06:52 AM
winterfell

Linear transformation of the line

So I have this question:

Find linear transformations of R2 which satisfy the following conditions: Sends the line y = 4x to the origina, but isn't the zero map.

I know that my teacher doesn't want us to use [0, 0, 0, 0] but rather find a transformation that sends the line back to the origin of the plane.

so y = 4x can become y - 4x = 0, so I would have to make both y and x 0 to make them collapse to the origin, right?
October 7th 2011, 10:16 AM
Deveno

Re: Linear transformation of the line

no. you would have to send all points (x,4x) to (0,0). it stands to reason that the expression y-4x should figure somewhere in the definition of your transformation.
October 7th 2011, 11:20 AM
HallsofIvy

Re: Linear transformation of the line

Any linear transformation from $R^2$ to $R^2$ can be written as
$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$
The fact that the entire line y= 4x is mapped to (0, 0) means that
$\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}x \\ 4x\end{bmatrix}= \begin{bmatrix}ax+ 4bx \\ cx+ 4dx\end{bmatrix}= \begin{bmatrix}0 \\ 0\end{bmatrix}$

which gives the two equations ax+ 4bx= 0 and cx+4dx= 0, for all x. Dividing through by x, a+ 4b= 0, c+ 4d= 0. Obviously a= b= c= d= 0 but that is not an acceptable soution. Instead write the equations as a= -4b, c= -4d. Choose any non-zero values you want for b and d.
October 7th 2011, 11:41 AM
Deveno

Re: Linear transformation of the line

i vote for b = d = 1, because 1 is my favoritest number in the hole-wide world.