Find Invariant Lines of Matrix Transformation y=mx+c

[Gregg]

Homework Statement

find in the form y= mx+c, the invariant lines of the tranformation with matrix

<br /> \left(<br /> \begin{array}{cc}<br /> 0 &amp; 1 \\<br /> 1 &amp; 0<br /> \end{array}<br /> \right)

<br /> <br /> \left(<br /> \begin{array}{cc}<br /> 0 &amp; 1 \\<br /> 1 &amp; 0<br /> \end{array}<br /> \right)\left(<br /> \begin{array}{c}<br /> x \\<br /> \text{mx}+c<br /> \end{array}<br /> \right)=\left(<br /> \begin{array}{c}<br /> \text{mx}+c \\<br /> x<br /> \end{array}<br /> \right)<br /> <br />

\Rightarrow x = m(mx+c)+c Why?

I just don't understand how that is implied in the first place and I don't have a method of working out invariant lines in the form mx or mx+c!

 

[rock.freak667]

Because you want a line whose x value will remain the same after undergoing the transformation.


So when you multiply the matrix by (x,y) you get (y,x). You then want your line to have the the x value of the 'old y value'

and if Y=MX+C

X= mx+c

so Y=M(mx+c) + C

(I used capital letters to explain it better even though, the capitals are the same as the common ones)