- #1

0kelvin

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## Homework Statement

I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is no orthogonal basis of eigenvectors for this particular linear transformation.

How do I show it?

The exercise doesn't ask this, but what's the implication of eigenvectors forming an orthogonal basis?

## Homework Equations

## The Attempt at a Solution

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With the euclidean inner product I can clearly see that the eigenvectors are not orthogonal to each other. But I'm not sure if calculating many pairs of dot products is the way to show it.

I thought about Gram-Schmidt but doing that would make the vectors not be eigenvectors anymore.

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