…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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