118.06 Quiz 2 April 7, 2010 Professor Strang Your PRINTED name is: 1. Your recitation number or instructor is 2. 3. 1. (30 points) (a) Find the matrix P that projects every vector b in R3 onto the line in the direction of a = (2, 1, 3). (b) What are the column space and nullspace of P ? Describe them geometrically and also give a basis for each space. (c) What are all the eigenvectors of P and their corresponding eigenvalues? (You can use the geometry of projections, not a messy calculation.) The diagonal entries of P add up to .2. (30 points) (a) p = Ax� is the vector in C(A) nearest to a given vector b. If A has independent columns, what equation determines x� ? What are all the vectors perpendicular to the error e = b − Ax� ? What goes wrong if the columns of A are dependent? (b) Suppose A = QR where Q has orthonormal columns and R is upper triangular invertible. Find x� and p in terms of Q and R and b (not A). (c) (Separate question) If q1 and q2 are any orthonormal vectors in R5 , give a formula for the projection p of any vector b onto the plane spanned by q1 and q2 (write p as a combination of q1 and q2).3. (40 points) This problem is about the n by n matrix An that has zeros on its main diagonal and all other entries equal to −1. In MATLAB An = eye (n) − ones (n). (a) Find the determinant of An. Here is a suggested approach: Start by adding all rows (except the last) to the last row, and then factoring out a constant. (You could check n = 3 to have a start on part b.) (b) For any invertible matrix A, the (1, 1) entry of A−1 is the ratio of . So the (1, 1) entry of A−41 is . (c) Find two orthogonal eigenvectors with A3 x = x. (So λ = 1 is a double eigenvalue.) (d) What is the third eigenvalue of A3 and a corresponding eigenvector?MIT OpenCourseWarehttp://ocw.mit.edu 18.06 Linear AlgebraSpring 2010 For information about citing these materials or our Terms of Use, visit:
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