MAT 2038 LINEAR ALGEBRA II 10.04.2014 Dokuz Eylül University, Faculty of Science, Department of Mathematics Instructors: Engin Mermut and Celal Cem Sarıoğlu web: http://kisi.deu.edu.tr/engin.mermut/ Course assistant: Zübeyir Türkoğlu SAMPLE QUIZ 3 • By answering the below questions, sketch the following quadric surface in x1 x2 x3 -space R3 : 2x1 x2 + 2x1 x3 + x22 + x23 = 2.   x1 − x =  x2  in R3 , 1. Find the SYMMETRIC 3 × 3 matrix A such that for all → x3 → − − x T A→ x = 2x1 x2 + 2x1 x3 + x22 + x23 . 2. What is the characteristic polynomial p(t) of the matrix A? p(t) = det(A − tI) =? 3. What are the eigenvalues of the matrix A? 4. For each eigenvalue λ of A, find an orthonormal basis for the eigenspace − − − E(λ) = {→ v ∈ R3 | A→ v = λ→ v} − − − 5. Combinining the orthonormal bases for each eigenspace, obtain an orhonormal basis B = {→ q 1, → q 2, → q 3} → − → − → − 3 for R such that q 1 , q 2 , q 3 are eigenvectors for A. 6. Find the 3 × 3 orthogonal matrix Q such that QT AQ = D is a diagonal matrix whose diagonal entries are the eigenvalues of A, counted with their algebraic multiplicity. − − − Hint: Let E = {→ e 1, → e 2, → e 3 } be the standard basis for R3 . Let µA : R3 → R3 be the linear transfor→ − − − mation defined by µ ( x ) = A→ x for all → x ∈ R3 . What are the matrices [µ ] and [µ ] , and what A E A A B is the relation between these matrices by the change of basis formula? Take Q to be the change of basis matrix PE←B . Since B is an orthonormal basis, observe that Q is an orthogonal matrix, that is, QT Q = I, and so Q−1 = QT . 7. Using the substitution     y1 x1 → − − y =  y2  = QT → x = QT  x2  y3 x3 the given equation and → − − x = Q→ y since Q−1 = QT → − − x T A→ x = 2x1 x2 + 2x1 x3 + x22 + x23 = 2 in x1 , x2 , x3 is transformed to the equation → − − y T D→ y = 2. in the variables y1 , y2 , y3 . Find this equation in the variables y1 , y2 , y3 , and sketch its graph in the y1 y2 y3 -space R3 . Is it an ellipsoid? Hyperboloid of one sheet? Hyperboloid of two sheets? Cylinder? What else? Then sketch the surface given in the beginning of the question in the x1 x2 x3 -space.