Suppose A is a square matrix, and let A, A2, ..., A be distinct eigenvalues of...

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Suppose A is a square matrix, and let A₁, A2, ..., A be distinct eigenvalues of A, with corresponding eigenvectors V1, V2, ... Vk. Suppose v₁ can be expressed as a linear combination of the remaining eigenvectors, V1 = A2V₂ + A3V3 + + akvk Prove: {V2, V3, ..., Vk} is a dependent set. Suggestion: What is Av₁? Edit Insert Formats === 2 BIU x₂ ײ A A e Prove: Eigenvectors for different eigenvalues must be linearly independent. Suggestion: If the eigenvectors are dependent, we can express one of them as a linear combination of the rest; the first part of this question shows that there will be a smaller set of dependent vectors. Lather, rinse, repeat. Edit Insert Formats B I U X₂ X² A = = E II e € A Σ+ Σ Α Edit Insert Formats BIU X₂ X² A A == E € Prove: An x n matrix can have at most n eigenvalues. DO NOT refer to any method of finding the eigenvalues. Σ+ Σ Α At 3+3 iin. Suppose A is a square matrix, and let A₁, A2, ..., A be distinct eigenvalues of A, with corresponding eigenvectors V1, V2, ... Vk. Suppose v₁ can be expressed as a linear combination of the remaining eigenvectors, V1 = A2V₂ + A3V3 + + akvk Prove: {V2, V3, ..., Vk} is a dependent set. Suggestion: What is Av₁? Edit Insert Formats === 2 BIU x₂ ײ A A e Prove: Eigenvectors for different eigenvalues must be linearly independent. Suggestion: If the eigenvectors are dependent, we can express one of them as a linear combination of the rest; the first part of this question shows that there will be a smaller set of dependent vectors. Lather, rinse, repeat. Edit Insert Formats B I U X₂ X² A = = E II e € A Σ+ Σ Α Edit Insert Formats BIU X₂ X² A A == E € Prove: An x n matrix can have at most n eigenvalues. DO NOT refer to any method of finding the eigenvalues. Σ+ Σ Α At 3+3 iin.