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sid9221
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So I'm a bit confused between these two and can't quite find any useful resources online. So is an eigenspace a special type of eigenvector cause that's how I understand it now.
An eigenspace is a vector space associated with a particular eigenvalue of a linear transformation. It consists of all the eigenvectors corresponding to that eigenvalue, along with the zero vector.
An eigenvector is a vector that, when multiplied by a linear transformation, results in a scalar multiple of itself. In other words, the direction of the eigenvector is not changed by the transformation, only its magnitude.
The main difference is that an eigenspace is a vector space, while an eigenvector is a single vector. An eigenspace contains all the eigenvectors associated with a specific eigenvalue, while an eigenvector is just one of those vectors.
To find an eigenspace, you first need to find the eigenvalues of a linear transformation. Then, for each eigenvalue, you can find the corresponding eigenvectors. The eigenspace is the vector space formed by these eigenvectors and the zero vector.
Eigenspaces and eigenvectors are important because they provide insight into the behavior of linear transformations. They can help us understand how a transformation affects different directions in a vector space and can be used to simplify calculations and solve problems in various fields of science, such as physics, engineering, and data analysis.