MATH 221 University of New Haven Gauss Jordan Elimination Exam Practice

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M 221 — Introduction to Linear Algebra — Fall 2020 Final Exam Due Tuesday, November 24, 2020, 11:59pm on Gradescope Closed book, show your work, write neatly. Good luck! Student Name: Question: 1 2 3 4 5 6 7 8 9 Total Points: 5 4 6 10 7 5 3 4 6 50 Score: 1. Read carefully: (a) (1 point) Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy. true false (b) (1 point) If, for some vector x, we have Ax = λx, then λ is an eigenvalue of A. (trick question!) true false (c) (1 point) Eigenvectors corresponding to distinct eigenvalues for a real symmetric matrix are always orthogonal. true false   −2 v= 6  1 (d) (1 point) The vector v is in the null space of A: true  A=  3 1 0 . −5 1 −4 false (e) (1 point) A positive definite matrix is always invertible. true false   1 2. (4 points) Suppose that a 2 × 2 matrix A has an eigenvalue 3 with corresponding eigenvector and an eigenvalue −1 2   1 with corresponding eigenvector . Find an invertible V and a diagonal D, such that A = V DV −1 : 1        A=    ·     ·      Page 1/6     1 −2 0 1 3. Consider the matrix B = −2 2 1 and the vector d = 0. 0 1 4 0 (a) (2 points) Compute the B = LU decomposition. (verify that B = LU !) L= U= (b) (1 point) Compute the determinant det(B). (Hint: not much work needed, now...) det(B) = (c) (3 points) Solve Bx = d using forward/backward substitution. x= Page 2/6     −1 2 1 1 and b = . 4. Consider the matrix A = −2 4 2 2     (a) (2 points) Perform elimination on A | b to get the reduced row echelon form R | c . R= c= (b) (2 points) Find the dimension of and give a basis for the column space C(A) dim(C(A)) = basis of C(A) =                                                                   (c) (2 points) Find the dimension of and give a basis for the null space N(A) dim(N(A)) = basis of N(A) = (d) (2 points) Find the dimension of and give a basis for the row space C(AT ) dim(C(A> )) = basis of C(A> ) = (e) (2 points) Give the complete solution of Ax = b. x∈ Page 3/6                         0 0 1 5. Consider the matrix C = 0 1 0. 1 0 2 (a) (1 point) Compute det(C). det(C) = (b) (1 point) “The matrix C is invertible.” true false (c) (1 point) λ1 = 1 is an eigenvalue of C. Find the associated eigenvector x1 . x1 = (d) (2 points) Find the other two eigenvalues. (Hint: find the characteristic polynomial and don’t simplify—polynomial division will be easy) λ2 = (e) (2 points) Find A−1 by Gauss-Jordan elimination A−1 = Page 4/6 λ3 =  1 −1 6. (5 points) (QR decomposition) Consider the matrix A =  1 −1  1 0 . 1 0 Find matrices Q (4 × 2) with orthonormal columns and R (2 × 2) upper triangular, such that A = QR. Q=  1 2 −1 1 7. (3 points) Let D =  0 1 1 −1 0 0 2 0 R=  2 2 . Compute det(D). 1 3 det(D) = 8. Linear independence. (a) (2 points) Complete the definition: “A set of n vectors v1 , v2 , . . . , vn is linearly independent if and only if . . .” Page 5/6       4 2   1 (b) (2 points) The set 2 , 5 , 1 is linearly   3 6 0 dependent independent Show your work:   2 4 9. Consider the matrix G = 1 2 (a) (2 points) Find eigenvalues λ1 > λ2 of G> G and their associated unit eigenvectors v1 and v2 λ1 = λ2 = v1 = v2 = (b) (2 points) Complete the SVD (e.g. using Avi = σi ui ):        G = UΣV > =     ·     ·      (c) (2 points) From its SVD, write down orthonormal bases for the four fundamental subspaces of G (give numbers!): basis C(G) =               basis N(G> ) =               basis C(G> ) = Page 6/6               basis N(G) =              
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