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What's a "trivial eigenspace"?
A trivial eigenspace is a special type of eigenspace that contains only the zero vector. This means that when a linear transformation is applied to the zero vector, the resulting vector is still the zero vector.
Unlike a regular eigenspace, which contains at least one non-zero vector, a trivial eigenspace only contains the zero vector. This means that the eigenvalue associated with the trivial eigenspace is always 0.
The existence of a trivial eigenspace for a linear transformation means that the transformation does not have any distinct eigenvectors. This can provide information about the nature of the transformation and its effect on vectors.
No, a linear transformation can only have one trivial eigenspace, which is the eigenspace associated with the eigenvalue of 0.
Trivial eigenspaces can be used to determine the eigenvalues and eigenvectors of a linear transformation. They can also provide information about the invertibility and diagonalizability of a matrix.