Hi, can someone help me get started on the following question?(adsbygoogle = window.adsbygoogle || []).push({});

Q. Show that there is no line in the real plane R^2 through the origin which is invariant under the transformation whose matrix is:

[tex]

A\left( \theta \right) = \left[ {\begin{array}{*{20}c}

{\cos \theta } & { - \sin \theta } \\

{\sin \theta } & {\cos \theta } \\

\end{array}} \right]

[/tex]

The problem is that I don't know what is meant by "invariant under the transformation." Another question asks for 1-dimesional subspaces of R^2 under the operation of the matrix [tex]\left[ {\begin{array}{*{20}c}

1 & 0 \\

2 & 0 \\

\end{array}} \right][/tex] .

The answer is span(0,1) and span(1,2). I don't know why those are the answers but with a little bit of guess work I decided to find the eigenspaces of the of the matrix and found that the two eigenspaces were exactly those answers. So is there some relationship between eigenspaces/ eigenvectors/ eigenvalues and invariance under transformations?

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# Linear Transformations

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