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9-6. Let ₁ and 2 be two eigenfunctions of a linear operator corresponding to the same eigenvalue. Show that any linear combination c₁₁ + C₂2 is also an eigenfunction with t same eigenvalue.

9-6. Let ₁ and 2 be two eigenfunctions of a linear operator corresponding to the same eigenvalue. Show that any linear combination c₁₁ + C₂2 is also an eigenfunction with t same eigenvalue.

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9-6. Let Y₁ and Y₂ be two eigenfunctions of a linear operator corresponding to the same eigenvalue. Show that any linear combination c₁Y₁+ C₂₂ is also an eigenfunction with the same eigenvalue. 9-7. In 9-6, ₁ and 2 may not be orthogonal (unlike two eigenfunctions of a Hermitian operator corresponding to two eigenvalues). Show, however, that one can always find an eigenfunction Y3 = C₁Y₁ + Y2, which is orthogonal to ₁. You may assume that ₁ and Y normalized.
Transcribed Image Text:9-6. Let Y₁ and Y₂ be two eigenfunctions of a linear operator corresponding to the same eigenvalue. Show that any linear combination c₁Y₁+ C₂₂ is also an eigenfunction with the same eigenvalue. 9-7. In 9-6, ₁ and 2 may not be orthogonal (unlike two eigenfunctions of a Hermitian operator corresponding to two eigenvalues). Show, however, that one can always find an eigenfunction Y3 = C₁Y₁ + Y2, which is orthogonal to ₁. You may assume that ₁ and Y normalized.
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