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## Homework Statement

Suppose the matrix

**A**with real entries has the complex eigenvalue λ=α+iβ, β does not equal 0. Let

**Y**be an eigenvector for λ and write

_{0}**Y**=

_{0}**Y**+i

_{1}**Y**, where

_{2}**Y**=(x

_{1}_{1}, y

_{1}) and

**Y**=(x

_{2}_{2}, y

_{2}) have real entries. Show that

**Y**and

_{1}**Y**are linearly independent.

_{2}[Hint: Suppose they are not linearly independent. Then (x

_{2}, y

_{2})=k(x

_{1}, y

_{1[/SUB) for some constant k. Then Y0=(1+ik)Y1. Then use the fact that Y0 is an eigenvector of A and that AY1 contains no imaginary part. Homework Equations AY=λY The Attempt at a Solution Honestly, not too sure where to start for this one. I know I should begin by considering the scenario where Y1 and Y2 are not linearly independent, but I do not know where I should begin with this information. Thanks for your help :)}