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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
Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.
The kernels of powers of a linear transformation form an ascending chain
The range spaces of powers of a linear transformation form a descending chain

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Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.

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