Find the equation of the invariant line through the origin

In summary, the equation of the invariant line through the origin can be determined by identifying a line that remains unchanged under a specific transformation or linear mapping. This typically involves finding a direction vector that satisfies the condition of the transformation, leading to a line equation in the form y = mx, where m is the slope determined by the transformation's characteristics.
  • #1
chwala
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Homework Statement
My interest is on highlighted in yellow. part b
Relevant Equations
see attached
1712742976617.png


My approach - i think similar to ms approach.

The required Equation will be in the form ##y=mx##

##\begin{pmatrix}
a & b^2 \\
c^2 & a
\end{pmatrix} ⋅
\begin{pmatrix}
k \\
mk
\end{pmatrix} =
\begin{pmatrix}
x \\
y
\end{pmatrix}
##



##ak+b^2mk=x##
##kc^2+amk=y##

##x=k(a+b^2m)##
##k=\dfrac{x}{a+b^2m}##

##y= k(c^2+am)##
##y=\dfrac{c^2+am}{a+b^2m}x##

##m=\dfrac{c^2+am}{a+b^2m}##

##am+b^2m^2=c^2+am##
##b^2m^2-c^2=0##
##m=\sqrt {\dfrac{c^2}{b^2}}##

##m_1 = \dfrac{c}{b}## and ##m_2 = -\dfrac {c}{b}##

##y=\dfrac{c}{b}x##

and

##y=-\dfrac{c}{b}x##

Ms approach,
1712743125853.png




Any insight welcome guys!
 
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  • #2
There's a general form for a matrix describing a rotation about the origin by an a gle ## \theta##.
You can derive it by seeing what happens when you rotate the point ##P=(cost, sint)## to the point ##P'=(cos(t+\theta), sin(t+\theta))##. Then expand the latter expression using the formulas for sin, cos of the sum of angles (show the map is linear), and use it to describe the matrix that takes you from ##P## to ##P'##.
Use that general form to test against the matrix you're given.
Can you take it from there?
 
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  • #3
WWGD said:
There's a general form for a matrix describing a rotation about the origin by an a gle ## \theta##.
You can derive it by seeing what happens when you rotate the point ##P=(cost, sint)## to the point ##P'=(cos(t+\theta), sin(t+\theta))##. Then expand the latter expression using the formulas for sin, cos of the sum of angles (show the map is linear), and use it to describe the matrix that takes you from ##P## to ##P'##.
Use that general form to test against the matrix you're given.
Can you take it from there?
I will need to check on this- self studying... thanks.
 
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FAQ: Find the equation of the invariant line through the origin

What is an invariant line?

An invariant line is a line that remains unchanged under a specific transformation or mapping. In the context of linear transformations, it refers to a line through the origin where any point on the line, when transformed, will still lie on that same line.

How do I find the equation of the invariant line through the origin?

To find the equation of the invariant line through the origin, you typically start with a linear transformation represented by a matrix. You can find the eigenvectors of the matrix associated with the transformation; the direction of these eigenvectors will give you the direction of the invariant line. The equation can then be expressed in the form \(y = mx\), where \(m\) is the slope determined by the eigenvector.

What role do eigenvalues and eigenvectors play in finding the invariant line?

Eigenvalues and eigenvectors are crucial in finding the invariant line. The eigenvectors corresponding to the eigenvalue of 1 represent the directions in which points remain unchanged under the transformation. Thus, the invariant line can be derived from these eigenvectors, indicating the lines through the origin that are preserved by the transformation.

Can there be more than one invariant line through the origin?

Yes, there can be multiple invariant lines through the origin, especially when dealing with transformations that have more than one eigenvalue of 1. In such cases, each eigenvector corresponding to the eigenvalue of 1 will define a distinct invariant line through the origin.

What happens if the transformation does not have any eigenvalues equal to 1?

If the transformation does not have any eigenvalues equal to 1, it means that there are no invariant lines through the origin for that transformation. In this case, all points will be transformed to other locations, and the origin will not remain invariant under the mapping.

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