How Do Eigenvectors Relate to Matrix Dimensions and Images?

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In summary, eigenvectors are vectors that, when multiplied by a matrix, result in a scaled version of themselves. They provide insight into the behavior of linear transformations and are important in solving systems of linear equations. To calculate eigenvectors, one must first find the eigenvalues of the matrix and then solve for the corresponding eigenvector. Eigenvectors and eigenvalues serve different purposes, with eigenvectors representing the direction of the transformation and eigenvalues representing the amount of scaling. In data analysis, eigenvectors are commonly used in techniques like principal component analysis (PCA) to identify important features and reduce the dimensionality of large datasets.
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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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What is an eigenvector?

An eigenvector is a vector that, when multiplied by a specific matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector remains the same, but its magnitude is scaled by a constant factor.

What is the significance of eigenvectors?

Eigenvectors are important because they provide insight into the behavior of linear transformations, such as stretching or compressing, in a given direction. They also play a crucial role in solving systems of linear equations and in understanding complex systems in fields such as physics and engineering.

How do you calculate eigenvectors?

To calculate eigenvectors, you first need to find the eigenvalues of the matrix. Then, for each eigenvalue, you can use a system of equations to solve for the corresponding eigenvector. Alternatively, you can use software such as MATLAB or Python's NumPy library to calculate eigenvectors.

What is the difference between eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are closely related, but they serve different purposes. Eigenvectors are the vectors that are scaled by a constant factor when multiplied by a matrix, while eigenvalues are the corresponding scalar factors. In other words, eigenvectors are the "directions" of the transformation, while eigenvalues are the "amounts" by which those directions are scaled.

How are eigenvectors used in data analysis?

Eigenvectors are commonly used in data analysis, specifically in techniques such as principal component analysis (PCA). In PCA, eigenvectors are used to identify the most important features or variables in a dataset, and to reduce the dimensionality of the data while retaining the most important information. This can be useful in visualizing and understanding large datasets.

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