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…Can you help me complete these two questions,thank you………..>………..6. (10 points) Let V be a finite dimensional inner product space over F, let T:
VV be a linear
operator, and let W be a subspace of V. Suppose that both W and w+ are T-invariant and
that the characteristic polynomial of T splits over F. Prove that there exists an orthonormal
basis ß of V such that
А 0
[TDB
O’ B
where O and O’ are matrices with only zero entries, and A and B are square matrices which
are upper-triangular.
Hint. Use Schur’s theorem, but remember you must verify the hypothises of the theorem.
4. (10 points) Let V be a finite dimensional inner product space and let V1,…, Un, w1,…, Wm,
21,…, & E V be distinct vectors. Let Si {V1, …, Vn}, S2 {wi, …, Wm), and S3
{x1,…,xl}. Assume that each of the sets S1, S2, and S3 are linearly independent, and assume
that for every 1
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