Hi.

A transformation $\displaystyle T$ : $\displaystyle R^2 \rightarrow R^2$ is represented by the matrix $\displaystyle A = \left(\begin{array}{cc}4&-5\\6&-9\end{array}\right)$.

There is a line through the origin for which every point is mapped onto itself under $\displaystyle T$.

Find the cartesian equation of this line.

I have already worked out the two eigenvalues for this matrix, 1 and -6.

Forming the equation:

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

Where $\displaystyle \lambda$ are my respective eigenvalues.

The question I have is which on do I use, using 1 I get $\displaystyle 5y = 3x$ and using -6 I get $\displaystyle y = 2x$.

My answers tell me that;

So 1 is the value I should use, but what exactly does the above expression mean?

Thanks in advance