- #1

Appleton

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## Homework Statement

Find the two numerical values of λ such that

[itex]

\left(\begin{array}{cc}4&3\\1&2\end{array}\right)

\left(\begin{array}{cc}u\\1\end{array}\right)

=λ

\left(\begin{array}{cc}u\\1\end{array}\right)

[/itex]

Hence or otherwise find the equations of the two lines through the origin which are invariant under the transformation of the plane defined by

[itex]

\left(\begin{array}{cc}x\prime\\y\prime\end{array}\right)

=

\left(\begin{array}{cc}4&3\\1&2\end{array}\right)

\left(\begin{array}{cc}x\\y\end{array}\right)

[/itex]

## Homework Equations

## The Attempt at a Solution

I believe [itex]

\left(\begin{array}{cc}4&3\\1&2\end{array}\right)

[/itex] represents a non uniform scale and shear. If λ is a numerical value it can only scale uniformly which suggests there is no solution to the initial equation. However, my textbook tells me the 2 numerical values are 5 and 1.

I skipped this first part of the question and found the two invariant lines by setting [itex]y=mx+c [/itex] equal to [itex] y\prime=mx\prime+c[/itex]. If someone could help me understand the first part of the question I would be very appreciative.