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Math 235 Final Exam Spring 1999

  1. (15 points) The matrices A and B below are row equivalent (you do not need to check this fact).

    tex2html_wrap_inline150 tex2html_wrap_inline152

    a) What is the rank of A?

    b) Find a basis for the null space Null(A) of A.

    c) Find a basis for the column space of A.

    d) Find a basis for the row space of A.

  2. (4 points) The null space of the tex2html_wrap_inline164 matrix A is 2 dimensional. What is the dimension of (a) the Row space of A? (b) the Column space of A? Justify your answer!
  3. (15 points)
    1. Show that the characteristic polynomial of the matrix tex2html_wrap_inline172 is tex2html_wrap_inline174 .
    2. Find a basis of tex2html_wrap_inline176 consisting of eigenvectors of A.
    3. Find an invertible matrix P and a diagonal matrix D such that the matrix A above satisfies

      displaymath144

  4. (12 points) Determine for which of the following matrices A below there exists an invertible matrix P (with real entries) such that tex2html_wrap_inline190 is a diagonal matrix. You do not need to find P. Justify your answer!
    1. tex2html_wrap_inline194
    2. tex2html_wrap_inline196
    3. tex2html_wrap_inline198
  5. (22 points) Let W be the plane in tex2html_wrap_inline176 spanned by tex2html_wrap_inline204 and tex2html_wrap_inline206
    Note: Parts 1, 2, 3 are mutually independent and are not needed for doing parts 4, 5, 6.
    1.   Find the length of tex2html_wrap_inline208 .
    2.   Find the distance between the two points tex2html_wrap_inline208 and tex2html_wrap_inline212 in tex2html_wrap_inline176 .
    3.   Find a vector of length 1 which is orthogonal to W.
    4.   Find the projection of tex2html_wrap_inline212 to the line spanned by tex2html_wrap_inline208 .
    5.   Write tex2html_wrap_inline212 as the sum of a vector parallel to tex2html_wrap_inline208 and a vector orthogonal to tex2html_wrap_inline208 .
    6.   Find an orthogonal basis for W.

  6. (16 points) Let W be the plane in tex2html_wrap_inline176 spanned by tex2html_wrap_inline236 and tex2html_wrap_inline238
    1. Find the projection of tex2html_wrap_inline240 to W.
    2. Find the distance from b to W.
    3. Find a least square solution to the equation Ax=b where A is the tex2html_wrap_inline252 matrix with columns tex2html_wrap_inline254 and tex2html_wrap_inline256 . I.e., find a vector x in tex2html_wrap_inline260 which minimizes the length tex2html_wrap_inline262 .
    4. Find the coefficients tex2html_wrap_inline264 , tex2html_wrap_inline266 of the line tex2html_wrap_inline268 which best fits the three points tex2html_wrap_inline270 , tex2html_wrap_inline272 , tex2html_wrap_inline274 in the x,y plane. The line should minimize the sum tex2html_wrap_inline278 . Justify your answer!

  7. (16 points) The vectors tex2html_wrap_inline280 and tex2html_wrap_inline282 are eigenvectors of the matrix tex2html_wrap_inline284 .
    1. The eigenvalue of tex2html_wrap_inline208 is ______

      The eigenvalue of tex2html_wrap_inline212 is _______

    2. Find the coordinates of tex2html_wrap_inline290 in the basis tex2html_wrap_inline292 .
    3. Compute tex2html_wrap_inline294 .
    4. As n gets larger, the vector tex2html_wrap_inline298 approaches _____. Justify your answer.




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Eyal Markman
Tue Nov 21 17:03:34 EST 2000