pfinal.pdf

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Department
Mathematics
Course
MATH 1502
Professor
Blekherman
Semester
Fall

Description
Math 1502 Practice Final Exam 1a. Find the eigenvalues and corresponding eigenvectors to the matrix −1 6 A = 6 4 b. Find A5(Show all work). c. Write the quadratic foTm Q associated with A and plot¯) = 2.rve Q(x d. Determine whether¯) =¯ Ax¯ is positive definite, negative definite, or indefinite. e. Show that if A is symmetric then the eigenvectors associated with different eigenvalues are orthogonal. f. Find det(A) where   1 2 2 1  1 0 1 1 A =   . 1 1 0 2 2 1 1 0 2.a Show whether or not the set of vectors       1 1 1 V = {¯ = t 2 + s 1 +  0 ,−∞ < t,s < ∞} 1 1 2 is a subspace. b. Given the vectors  1   1   1 ¯1=  1  ¯2=  0  ¯3=  0 , −2 −1 1 0 1 1 find an orthonormal basis for V¯1¯2,3 ). 1 1  c. Find the orthogonal projec¯ =nofxin V from part b. 1 4 3. (a) Find a basis for colA and NulA of the following matrix.   1 1 −1 0 −1 1 0 1 1 0  A =   1 1 −1 1 −2 1 0 1 1 0 (b) Find the dim(Nul A) and rankA T T (c) Find a basis for the and col A . 1 2 1 4a. Find and graph all the solutions to the equat¯ = b where A = , and 0
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