MATH 2270-003 Exam 2 Fall 2010NAME :You may not use a calculator. Your solutions must include enough justification that an-other person could understand and be convinced by your argument.There are extra blank pages at the end of the booklet. If you need more room to work aproblem please note the page number where your work continues.QUESTION VALUE SCORE1 102 153 154 155 56 57 158 59 1010 15TOTAL 11011. (10 points) V is the span of the given vectors in R4. Find orthonormal vectors whosespan is V .¯v1=0−120, ¯v2=3113√222. (15 points) For the subspace V in the previous problem, give the matrix that projectsR4to V and the matrix that projects R4to V⊥.33. (15 points) Find the least squares best fit line for the points (0, 1), (1, 2), (2, 3), (4, 4).44. (15 points) For the following matrix, find the eigenvalues and the maximum numberof linearly independent eigenvectors. Find this many linearly independent eigenvectors.A = 4 1−5 −2!55. (5 points) For the following matrix, find the eigenvalues and the maximum numberof linearly independent eigenvectors. Find this many linearly independent eigenvectors.A =1 0 0 0 0 00 2 1 0 0 00 0 2 0 0 00 0 0 3 0 00 0 0 0 3 10 0 0 0 0 36. (5 points) Find the determinant of the following matrix:3 2 31 0 10 1 −167. (15 points) Describe the orbits of the discrete linear dynamical system ¯vi+1= A¯viforthe matrixA = 0 −12 3!78. (5 points) Suppose V is a 4 dimensional subspace of R9. Let PVbe the matrix thatprojects R9onto the subspace V . What are the dimensions and rank of the matrix PV?9. (10 points) Suppose A is a 3 × 3 matrix whose entries all have absolute value lessthan or equal to 2. Find such a matrix that has Det(A) ≥ 30. Is it possible to find such amatrix with Det(A) ≥ 50? Find one or explain why it is impossible.810. (15 points)A =2 0 00 4 20 2 1Find a matrix C such that CAC−1is diagonal.No new questions beyond this
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