How Do Eigenvectors Relate to Matrix Dimensions and Images?

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SUMMARY

The discussion focuses on the relationship between eigenvectors and matrix dimensions, specifically in the context of 2x2 matrices. It establishes that for any nonzero vector v in R^2, the dimension of the space V of all 2x2 matrices for which v is an eigenvector is 3. Additionally, it confirms that if v is an eigenvector of matrix A with an associated eigenvalue of 3, then v is indeed in the image of matrix A, as defined by the equation Av = cv.

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  • Understanding of eigenvectors and eigenvalues in linear algebra
  • Familiarity with matrix dimensions and their properties
  • Knowledge of the definition and implications of the image of a matrix
  • Basic proficiency in R^2 vector space concepts
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Homework Statement



1. If v is any nonzero vector in R^2, what is the dimension of the space V of all 2x2 matrices for which v is an eigenvector?

2. If v is an eigenvector of matrix A with associated eigenvalue 3, show that v is in the image of matrix A

Homework Equations



If v is an eigenvector with eigenvalue c(real number), then Av=cv (definition of eigenvector)

The Attempt at a Solution



i have posted a picture for my attempt at the first question
but i totally have no idea on the second question
need help from you guys!
 
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