If U is invariant under T, then U is invariant under T2 2. If U is invariant under T2, then U is...

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If U is invariant under T, then U is invariant under T2 2. If U is invariant under T2, then U is invariant under T 3. Suppose that U is a 1-dimensional invariant subspace of T. Then T has at least one eigenvalue (the field of scalars is R) 4. There exists T : R3 —. R3 with eigenvalues {1, Z 3,4} 5. If dim (V) > 0, then T : V —. V has at least 2 invariant subspaces 6. Let p (z) be a polynomial. Then if 7 is an eigenvalue of T, then p(7) is an eigenvalue of p(T). 7. Every T :R2 —. R2 has an upper triangular representation. 8. Let p(s) be a non-trivial polynomial. Then if p(T) has an eigenvalue, so does T (the field is R). 9. Every T : C2 -. C2 has an upper triangular representation. 10. Suppose that 3 is an eigenvalue ofT:V —. V and that S is invertible. Then 3 is also an eigenvalue of S ITS. 11. If T : Whits an upper triangular representation then T has at least one eigenvector. 12. If [0 3I 21 is a matrix representation of T : V —. V then T has two linearly independent eigenvectors. 13. Suppose that T : P3 -. P3 is linear. Then there exists a non-zero polyno- mial p E P3 such that {p, Tp,T2p, 7.3p, 74p} is linearly independent. 14. Every T C' has an invariant subspace of dimension XXXXXXXXXXIf T : M253 -• Af2x3 has a diagonal representation, then T has at least 6 linearly independent eigenvectors. 16. If T : 1112.3 —• M3,3 has an upper triangular representation, then T has at least 6 linearly independent eigenvectors.

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Robert · Accepted Answer

Clarification
13. Yes you are right and it is trivial that it can not be basis. I misunderstood
as P4.. But it is P3.
Hence the set can never be linearly independent.
14. Take linear operator such that matrix representation of T is such that
1 1 0 ... 0
0 1 1 ... 0
0 0 1 ... 0
. . . ... .
0 0 0 ... 1
0 0 0 ...

If U is invariant under T, then U is invariant under T2 2. If U is invariant under T2, then U is invariant under T 3. Suppose that U is a 1-dimensional invariant subspace of T. Then T has at least one...

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