Math Help - Matrix Transformation problem

1. Matrix Transformation problem

Hi, another question from me again

A transformation $T$ : $R^2 \rightarrow R^2$ is represented by the matrix $A = \left(\begin{array}{cc}4&4\\4&-2\end{array}\right)$.

Find:

a) The two eigen values for A
b) A cartesian equation for each of the two lines passing through the origin which are invariant under $T$.

a) This was easy enough, -4 and 6.

b) No idea how to start this, anyone got any pointers?

Thanks

2. Invariant means that it does not change under the transformation doesn't it? I'm guessing that you use eigenvalues and eigenvectors at some point?

3. Sorry just one more thing.

Could someone please explain what exactly is meant by:

$R^2 \rightarrow R^2$
Not too sure what this means :S

Thanks again

4. Originally Posted by craig
Hi, another question from me again

A transformation $T$ : $R^2 \rightarrow R^2$ is represented by the matrix $A = \left(\begin{array}{cc}4&4\\4&-2\end{array}\right)$.

Find:

a) The two eigen values for A
b) A cartesian equation for each of the two lines passing through the origin which are invariant under $T$.

a) This was easy enough, -4 and 6.

b) No idea how to start this, anyone got any pointers?

Thanks
Originally Posted by craig
Invariant means that it does not change under the transformation doesn't it? I'm guessing that you use eigenvalues and eigenvectors at some point?
It means that points on that line are mapped into points on that line. For eigenvectors, v, [tex]Av= \lamba v[/itex] and, interpreting v as position vector that means Av points in the same direction as v: They lie on the same line as v.

Yes, part b is just asking you to find the eigenvectors corresponding to the eigenvalues and then write the equations of line through the origin in the direction of those vectors. And that's easy: The fact that -4 is an eigenvalue means that there exist non-zero x, y satisfying $\begin{bmatrix}4 & 4 \\ 4 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}4x \\ 4y\end{bmatrix}$. Multiplying that out and setting components equal immediately gives two equations that both satisfy y= -2x. That is one of the invariant lines.

5. Thanks for the reply, could you just verify this for me if possible?

Originally Posted by HallsofIvy
Yes, part b is just asking you to find the eigenvectors corresponding to the eigenvalues and then write the equations of line through the origin in the direction of those vectors.
Using -4:

$\left(\begin{array}{cc}4&4\\4&-2\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = -4\left(\begin{array}{cc}x\\y\end{array}\right)$

$4x + 4y = -4x \rightarrow y = -2x$

And using 6

$\left(\begin{array}{cc}4&4\\4&-2\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = 6\left(\begin{array}{cc}x\\y\end{array}\right)$

$4x + 4y = 6x \rightarrow y = \frac{1}{2}x$

Would this be correct?