Two eigenvectors corresponding to the same eigenvalue must be linear dependent.
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I think it is true.
No, it's false. For example, the identity matrix $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ has 1 as its only eigenvalue. Both [1 0] and [0 1] are eigenvectors corresponding to eigenvalue 1 and are independent.
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