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Exercise 7.5.2: Evaluating compositions. Let Ti : R3 - R3, T2 : R' _ R3, and T3 : R -> R' be linear transformations defined as follows. 1+2 T1 2+3 T2 = 21 - $2 T3 2 C1 - 22 C3 Find each composition, if the composition exists. (a (T2 0 71) 0 (b) (T2 0 T3) () (C) (Ti O T2) ( 3])Exercise 7.5.3: Matrix representation of a composition. Let 71 : R' -> R4 and 72 : R4 - R be linear transformations defined as follows. 2x1 1 2 T1 C1 = C1 - $2 1 + 2 = T2 3x2 E3 + 4 La1 + 5x2 Find the matrix representation for the composition given. (a) T2 o T1 (b) TioT2Exercise 7.5.4: Inverse linear transformations. Determine if each transformation is invertible. If so, give the matrix representation for the inverse transformation. (a) T : R' -> R given by T x1+ 2x2 232 (b) 2x2 - 3x3 T : R3 - R given by T = 4x2 + 6x3 T3 1 + 3x2 - 203Exercise 7.5.5: Finding the inverse of a composition of linear functions. Let 71 : R' -> R' and 72 : R' -> R be the invertible linear transformations defined as follows. T1 ([ = ]) 21 + 3x2 T2 2x1 + 5x2 -3x1 - 5x2 OC 2 3x1 + 702 Find the matrix representation for the inverse composition transformation given. (a (T2 o Ti) -1 (b) (TioT2)-1Exercise 7.6.2: Finding the matrix representation of a linear transformation with respect to nonstandard bases. Find [T']g for each of the given linear transformations and bases B and C. (a) T : R3 - R' defined by T 1 + 202 -203 O B = (b) T : P3 - R defined by a1 + 3a3 T (a1x3 + a2x2 + agx + a4) = a2 -5a4 B = {1, x, x + 2, 1+23); C= (c) T : P1 + P2 defined by T(aix + ao) = (a1 + 2ao)x2 - (a1 + ao)x + 300 B = {1, 1-x} , C = {1, x + 2,x2 - 1}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