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Unformatted text preview: Review Questions for Midterm Exam
1. Know the ﬂoatingpoint number systems. All problems in the four homework sets we have
had so far are fair questions to ask on the exam. The exam will cover the three chapters
we have covered so far: ﬂoatingpoint number systems, linear leastsquares and Eigenvalues/Eigenvectors. The chapter on rootﬁnding will NOT be covered on the midterm exam.
2. • Let x be the solution to the linear least squares problem Ax 10
1 1 A=
1 2 .
13 b, where Let r = b − Ax be the corresponding residual vector. Which of the following three vectors
is a possible value for r? −1
−1
1
1
−1
1 . , ,
1
1
1
−1
1
1
• What is the Euclidean norm of the minimum residual vector for the following linear least
squares problems? 11
2
0 1 x1
1
x2
00
1
• Consider the vector a as an n × 1 matrix. What is the QR factorization of a? and What
is the solution to the linear least squares problem ax = b, where b is a given nvector?
3. Determine the Householder transformation that annihilates all but the ﬁrst entry of the vector
[1 1 1 1 ] . Speciﬁcally, if α
1
1 0 vv
I−2
= = v v 1 0 0
1
what are the values of α and v ?
4. Let A be a matrix with the following Singular Value Decomposision
A = [u1 ... um ] Σ [v1 ... vn ] .
How would you use this SVD to solve the linear least squares problem Ax b? 5. Let u, v be two vectors in Rn such that u v = 0.1. Consider the matrix A = uv .
• Is the matrix A diagonalizable?
• What are the eigenvalues of A?
• If power iteration is applied to A, how many iterations are required for it converge exactly
to the eigenvector corresponding to the dominant eigenvalue?
• Consider the matrix B = In + A, where In is the n × n identity matrix. What are the
trace and determinant of B ?
1 6. Eigenvalues and Eigenvectors 2 7. Eigenvalues and Eigenvectors 3 ...
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 Spring '08
 Davis
 Linear Algebra, Singular value decomposition, Orthogonal matrix, Linear least squares, QR algorithm, ﬂoatingpoint number systems

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