CAUSE
One question. One truth.
“Why T-cyclic subspace generated by a nonzero element is T-invariant?”
The T-cyclic subspace generated by a nonzero element v is T-invariant because applying T to any element p(T)v, where p(T) is a polynomial in T, results in (Tp(T))v, which remains in the subspace since Tp(T) is also a polynomial in T.
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CURRENT STATE
The T-cyclic subspace is defined as the span of vectors {v, T(v), T^2(v), ...}, and any element in it can be expressed as a polynomial in T applied to v. Applying T to such an element shifts the polynomial degree by one, keeping the result within the same span, thus preserving T-invariance.
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