Please help me find the answers to the practice question and please provide work,

Exercise 7.6.4: Understanding the Fundamental Theorem of Matrix Representations. Consider the linear transformation : R* -> P, defined by a1 T CL2 = 2a123 - (a1 + as)2 + 302 a3 Let u = O (a) Find Pc (T(u) ) (b) Find [TI (PB (u)) (c) Use the transformation matrix from the standard basis of R to the basis C of P3 to find T 0Exercise 7.6.5: Applying the Fundamental Theorem of Matrix Representations. Consider the linear transformation T' : R -> R* defined by -201 T a1 - a2 a2 a1 + a2 Let u = a Find T 6 for the standard basis of R and basis C of IR*Exercise 8.1.1: Eigenvalues and eigenvectors. Determine whether the statement is true or false. Justify each answer or provide a counterexample when appropriate. (a) For A ( Rnxn, X E R", and ) E R, (1, x) is an eigenpair of the matrix A if Ax = Ax and x / 0. (b) Eigenvalues can only be found for a square matrix. (c) The zero vector x = O ...0 0 is an eigenvector of every n X n matrix. (d) If () - 2) is a factor of the characteristic polynomial of A, then ) = -2 is an eigenvalue of A (e) If the characteristic equation of A is 1 - 51 + 6 = 0, then A is invertible. (f) For an n x n matrix A, the determinant of A is always equal to the product of the eigenvalues. (g) The spectrum of any linear transformation (x) = Ax is given by the spectrum of the standard matrix A.Exercise 8.1.2: Finding eigenvalues and eigenvectors of a matrix. Find the eigenvalues and eigenvectors as indicated. (a) -4 Find the eigenvalues of A = 0 1 0 1 0 1 (b) Find all eigenvalues and eigenvectors for A = 7 (c) 01 0 20 1 Find the eigenvector x corresponding to * = 2 for the matrix A = 1 10 0 OExercise 8.1.3: Finding eigenvalues and eigenvectors of a linear transformation. Find the eigenvalues of each transformation, if possible. (a) a T : P2 - R given by T (azz + bx + c) = 2a + 7b -a - 3b + 3c] (b) T : R3 - R' given by T 521 + 23 2 + 6x3