extra2 - × 4 where you can have orthogonal matrices which...

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Name: Linear Algebra Extra Credit for Final Exam June 13, 2006 The following problems may be done for extra credit on the final exam. More will be added as time goes on. These problems will be due on Friday June 23 at 5PM. Same policy as last time–please hand in original solutions. 1. (2 points) Let V = C [ - 1 , 1], that is continuous functions on the interval [ - 1 , 1]. Suppose that X e and X o denote the subspaces of V consisting of even and odd functions, respectfully. Show that X e = X o where V is equipped with the inner product h f,g i = Z 1 - 1 f ( t ) g ( t ) dt . 2. (2 points) Show that any 3 × 3 orthogonal matrix with determinant 1 has eigenvalue 1. This is useful in showing that 3 × 3 special orthogonal matrices (special=determinant 1) are nothing more than rotations about some axis in 3 space. Since the product of two orthogonal matrices is again orthogonal, this shows that the composition of two rotations in 3 space is again a rotation about some axis–a far from obvious geometric fact. Things get more complicated in 4
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Unformatted text preview: × 4, where you can have orthogonal matrices which are simultane-ously rotations about different axes, which cannot be written as a rotation about a single axis. 3. (3 points) Show that every matrix A ∈ M n × n with determinant 1 has a unique fac-torization A = RH where R is an orthogonal matrix with determinant 1 (ie, R T = R-1 ) and H is a symmetric, positive-definite matrix with determinant 1 (that is, H T = H and h x,Hx i ≥ 0 for all x ∈ R n with equality if and only if x = 0). You may assume that ev-ery real symmetric matrix is real diagonalizable–a fact which we will hopefully discuss in class. 4. (3 points) Consider R n with its standard inner product. Let U and T be symmetric n × n matrices. Suppose that UT = TU . Show that there exists an orthonormal basis for R n consisting of eigenvectors for both U and T . In other words, we may simultaneously diagonalize U and T . 1...
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