NAME:1 /20 2 /15 3 /17 4 /10 5 /10 6 /67 /8 8 /8 9 /6 10 /6 11 /14 T /120MATH 430 (Fall 2008) Final Exam 2, Dec 12thNo calculators, books or notes! Show all work and give complete explanations.This 75 minute exam is worth a total of 75 points.(1) [20 pts](a) Define the spectrum of a n × n matrix.(b) Let V be a finite dimensional vector space and let B be a basis for V. Define the matrix [T ]Bof alinear transformation T : V → V. Suppose that B0is another basis for V. How, precisely, are [T ]Band[T ]B0related?(c) State three properties that characterize the determinant of a square matrix.(d) Define the algebraic multiplicity and the geometric multiplicity of an eigenvalue. Which is larger?What can you conclude if all the eigenvalues of a matrix have algebraic multiplicity equal to 1?(e) Carefully state the version of the Spectral Theorem for diagonalizable matrices that involves spectralprojectors. (This result is sometimes called the Spectral Decomposition Theorem.)2(2) [15 pts] Let A be the matrixA =0 1 31 4 23 −1 5.(a) Calculate det(A) using row operations.(b) Calculate det(A) using a cofactor expansion.3(c) Let x = [x1, x2, x3]Tbe the s olution of Ax = b, where A is given above and b = [0, 3, −4]T. UseCramer’s Rule to calculate x2.(3) [17 pts] Suppose that A is a 3 × 3 matrix with eigenvalues λ1= 2 and λ2= 3 and eigenspacesN (A − 2I) = Span101,010N (A − 3I) = Span10−1.(a) Show that the function f : R3→ R defined by f(x) = xTAx is positive for all x 6= 0.4(b) Calculate the spectral projectors G1and G2corresponding to λ1and λ2.(c) Use (b) to solve the system of differential equationsdudt= Au, with initial condition u(0) = (1, 2, 3)T.5(4) [10 pts] Use least squares to find the best linear fit to the data (xi, yi) = (1, 2), (3, 5), (5, 7).6(5) [10 pts] Let V be the vector space that is spanned by the linearly independent functions p0(x) = 1,p1(x) = x, p2(x) = x2, p3(x) = x3. Find the eigenvalues of the linear transformationddx: V → V definedbyddx(f) =dfdx. Is there a basis B for V so thatddx Bis diagonal?7(6) [6 pts] Prove that the columns of an m×n matrix A are linearly independent if and only if N (A) = {0}.(7) [8 pts] Prove that λ is an eigenvalue of A if and only if det(A − λI) = 0.8(8) [8 pts] Let P be an orthogonal matrix. Prove that det(P) = ±1. Also, give an example of an orthogonalmatrix with det(P) = −1.(9) [6 pts] Let c and d be two non-zero n × 1 vectors. C alculate the rank of the matrix cdT.9(10) [6 pts] Let T : Rn→ R be a linear transformation. Find a vector u so that T(v) = uTv for allv ∈ Rn. Hint: Express v in the standard basis for Rn.(11) [14 pts] Let A be an m × n matrix with complex entries.(a) Prove that R(A)⊥= N (A∗).10(b) Prove that R(A∗) ⊆ N (A)⊥.(c) Using (a) and (b) prove that R(A∗) = N (A)⊥.Pledge: I have neither given nor received aid on this