The discussion confirms that a set of eigenvectors corresponding to different eigenvalues is linearly independent. This is established by the principle that if any two vectors in a set are independent, then all vectors in that set are independent. The proof involves demonstrating that if a linear combination of independent vectors remains independent of another vector, then the entire set maintains linear independence. This foundational concept is critical in linear algebra and has implications in various applications such as systems of differential equations and quantum mechanics.
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Do you mean that two eigenvectors corresponding to two different eigenvaluesFermat said:I know eigenvectors corresponding to different eigenvalues are linearly independent
but asking, "what if there are more than two?". One can show generally, "if, in a set of vectors, any two are independent (au+ bv= 0 only if a= b= 0 which is the same as saying that b is NOT a multiple of a and vice-versa) then all the vectors are independent."but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?