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

$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$

$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\left( \begin{array}{c} x \\ \text{mx}+c \end{array} \right)=\left( \begin{array}{c} \text{mx}+c \\ x \end{array} \right)$

$\Rightarrow x = m(mx+c)+c$ 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)

 

[Architeuthis Dux]

The term "invariant lines" refers to lines that do not change when the transformation is applied. In other words, the points on these lines are mapped to themselves by the transformation. In the given matrix transformation, we are looking for lines that remain unchanged when multiplied by the matrix. Invariant lines can be found by solving for the points that satisfy the equation:

\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right)\left(
\begin{array}{c}
x \\
\text{mx}+c
\end{array}
\right)=\left(
\begin{array}{c}
x \\
\text{mx}+c
\end{array}
\right)

This can be rewritten as:

\begin{align*}
0x + 1(\text{mx}+c) &= x \\
\text{mx} + c &= x \\
\Rightarrow x &= m(mx+c)+c
\end{align*}

This is the equation for the invariant lines in the form y=mx+c. To find these lines, you can use any method of solving linear equations, such as substitution or elimination. Alternatively, you can also graph the transformation and look for lines that remain unchanged after the transformation is applied.