NAME:1 /30 2 /10 3 /10 4 /8 5 /106 /10 7 /16 8 /10 9 /8 10 /8 T /120MATH 430 (Fall 2005) Final Exam, December 20thShow all work and give complete explanations for all your answers.This is a 120 minute exam. It is worth a total of 120 points.(1) [30 pts](a) Define the term maximal linearly independent set(b) Use matrix analysis to answer the question: Is it possible for the intersection of three planes to consistof a pair of points?(c) Define the algebraic and geometric multiplicities of an eigenvalue of a matrix A. Which is bigger?(d) State three conditions each of which guarantee that X and Y are complementary subspaces of a vectorspace V.2(e) Write down an equation that characterizes the adjoint T∗: V → V of a linear transformationT : V → V on an inner product space V. State how the matrix of T∗is related to the matrix of T.(f) State the Range-Nullspace Decomposition Theorem for Normal Matrices.3(2) [10 pts] Given that we know how to define the determinant of a matrix, define the determinant of alinear transformation T : V → V and explain why it is is well defined.(3) [10 pts] Prove that the eigenvalues of a Hermitian matrix are real.4(4) [8 pts] Define the coordinates of a vector u with respect to a basis B = {v1, v2, . . . , vn} and prove thatthey are unique.5(5) [10 pts] Prove that if a matrix A has a basis of eigenvectors then A is diagonalizable.6(6) [10 pts] Let A be a diagonalizable matrix. Show that cos2A + sin2A = I.7(7) [16 pts] Solve the initial value problemdu1dt= u2du2dt= 3u1+ 2u3du3dt= −u2with initial condition (u1, u2, u3)T= (1, 2, 3)Tat t = 0.8(7) This page is left blank deliberately to give more space for (7)9(8) [10 pts] Suppose that A is a 3 × 3 matrix with eigenvalues λ1= 2 and λ2= 3 and eigenspacesN(A −λ1I) = span√211andN(A −λ2I) = span01−1,−√211.Explain why A is a real symmetric matrix and write down the spectral decomposition of A (i.e., thedecomposition of A in terms of its spectral projectors).10(9) [8 pts] Use the Gram-Schmidt process to construct an orthonormal basis for the subspace of R3consisting of all vectors that are orthogonal to (1, −1, −2)T.11(10) [8 pts] Suppose that AT= −A is antisymmetric.(a) Prove that the matrix I − A is invertibleProve that the matrix Q = (I − A)−1(I + A) is orthogonal.Pledge: I have neither given nor received aid on this