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Math 2890 Sample Exam Problems

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Exam IV

6.1      Inner Product, Length, and Orthogonality
6.2      Orthogonal Sets
6.3      Orthogonal Projections
6.4      The Gram-Schmidt Process
         Wedderburn Rank Reduction
6.5      Least-Squares Problems
6.6      Applications to Linear Models

7.1      Diagonalization of Symmetric Matrices
7.2      Quadratic Forms
7.3      Constrained Optimization
7.4      The Singular Value Decomposition
7.5      Applications to Image Processing and Statistics
         The Great Human Diasporas
         Text Retrieval Algorithms
  1. Compute the inner products of two vectors.
  2. Find a unit vector in a given direction.
  3. Find the distance between two vectors.
  4. Determine if a set is orthogonal.
  5. Project `y` onto the subspace `W`.
  6. Find the closest point to `y` in the subspace `W`
  7. Write `y` as a sum of a vector in `W` and a vector orthgonal to `W`.
  8. Find the least squares solution of the system `Ax=b.`
  9. Find the least squares error in the least squares solution of `Ax=b.`
  10. Use the QR factorization of a matrix.
  11. Find the QR factorization of a matrix.
  12. Use the QDR factorization of a matrix.
  13. Find the QDR factorization of a matrix.
  14. Find the equation of the best least-squares approximation given some data points
  15. Orthogonally diagonalize a matrix.
  16. Find the matrix of a quadratic form.
  17. Compute the quadratic form `Q(x)=x^T Ax` given `A.`
  18. Maximize `x^TAx` subject to `x^Tx=1.`
  19. Find vector at which `x^TAx` is maximized subject to `x^Tx=1`
  20. Use the singular value decomposition of a matrix.
  21. Find the singular value decomposition of a matrix.
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