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saqifriends
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saqifriends said:https://www.physicsforums.com/attachments/215
The basis for each eigenspace is a set of linearly independent vectors that span the corresponding eigenspace.
To find the basis for an eigenspace, you first need to find all the eigenvectors corresponding to a specific eigenvalue. Then, you can use these eigenvectors to form a basis for the eigenspace by checking for linear independence and spanning the space.
Yes, the basis for an eigenspace can change depending on the chosen eigenvalue. Different eigenvalues may have different corresponding eigenvectors, thus resulting in a different basis for each eigenspace.
The basis for each eigenspace is important because it helps us understand the structure and behavior of a linear transformation. It also allows us to easily compute the eigenvalues and eigenvectors of a matrix.
The dimension of an eigenspace is equal to the number of vectors in the basis for that eigenspace. In other words, the basis for each eigenspace forms a basis for the entire eigenspace, and the number of vectors in the basis is the dimension of that eigenspace.