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Subspaces associated to a linear transformation

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The kernels of powers of a linear transformation form an ascending chain
The range/image of a linear transformation is a subspace.
The inverse image of a subspace under a linear transformation is a subspace.
Definition of kernel/null space of linear transformation
The kernel/null space of a linear transformation is a subspace
Definition of generalized kernel/null space of linear transformation
Finite-dimensional Fitting decomposition into generalized kernel and range
The range spaces of powers of a linear transformation form a descending chain

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The inverse image of a subspace under a linear transformation is a subspace.

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