Provide solutions. to the following attachments.

The budget constraint we considered thus far assumes only lump-sum taxes. In reality, lump-sum taxation is used very rarely. One of the reasons is that people are not identical and so the requirement that all people pay the same amount is not feasible. (Some people earn nothing and so are not able to pay any positive amount.) An alternative policy is a proportional labor income tax, where taxes are equal to a fixed fraction t, 0' for some vEV . d) Prove: Let I be an inner-product space, and let v and w be vectors in . Then Kv. w) =|~|wl. e) Prove: Let A be an axn matrix and let 4, 4,....A. be (possibly complex) scalars and V,V,:..V. be nonzero vectors in -space. Let C be the axe matrix having v, asjith column vector, and let D= . Then AC - CD if and only if 2. ...A. are 0 eigenvalues of A and v, is an cigenvector of A corresponding to A, for / = 1,2.. .. () Prove: Let A be an # xn matrix and let v, V,,...>. be eigenvectors of A corresponding to distiner eigenvalues A. A....A . respectively. the set fo, v....v. } is linearly independent and A is diagonalizable. Prove: If A is a Hermitian matrix, there exists a unitary matrix ( such that U" AU is a diagonal matrix and all eigenvalues of A are real. [ complete the proof started below] Proof. By Schur's lemma, there exists a unitary matrix ( such that U" AL is upper-triangular matrix. Because U is unitary, we have U"U = J . so U" = U" and because A is Hermitian, we also know Thus, we have which shows that the upper-triangular matrix CAU is also Hermitian. Because the conjugate transpose of an upper-triangular matrix is a lower-triangular matrix. we see that the entries above the diagonal in U" AU must all be zero; therefore. U AU - . Where is matrix. Thus, A is unitarily diagonalizable. Now we will show that each eigenvalues of A is a real number. [Finish this part of the proof] by Prove: The eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are orthogonal. Proof: Let v and w be eigenvectors of a Hermitian matrix 4 corresponding to distinct eigenvalues 4, and 2. respectively. Using the fact that Am " and that the eigenvalues are real, so that by = 2. we have 2 (wv)- ... =(wv). [Fill in details]. Therefore, (2, - 2, )(wv)- 0. [Finish proof]. Since the eigenvalues are distinct, specifically