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Math 235 Final Exam Fall 2000

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

    tex2html_wrap_inline147 tex2html_wrap_inline149

    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. (6 points) The system tex2html_wrap_inline161 has a 2-dimensional space of solutions and the size of the matrix A is tex2html_wrap_inline167 . What is the dimension of (a) the Null space of A? (b) the Column space of A? (c) the Row space of A? Justify your answers!
  3. (15 points)
    1. Show that the characteristic polynomial of the matrix tex2html_wrap_inline175 is tex2html_wrap_inline177 .
    2. Find a basis of tex2html_wrap_inline179 consisting of eigenvectors of A.
    3. Find an invertible matrix P and a diagonal matrix D such that the matrix A above satisfies


  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_inline193 is a diagonal matrix. You do not need to find P. Justify your answers!
    1. tex2html_wrap_inline197
    2. tex2html_wrap_inline199
    3. tex2html_wrap_inline201
  5. (20 points) Let W be the plane in tex2html_wrap_inline179 spanned by tex2html_wrap_inline207 and tex2html_wrap_inline209
    Note: Parts 5a, 5b are mutually independent and are not needed for doing parts 5c, 5d, 5e.
    1.   Find the distance between the two points tex2html_wrap_inline211 and tex2html_wrap_inline213 in tex2html_wrap_inline179 .
    2.   Find a vector of length 1 which is orthogonal to W.
    3.   Find the projection of tex2html_wrap_inline213 to the line spanned by tex2html_wrap_inline211 .
    4.   Write tex2html_wrap_inline213 as the sum of a vector parallel to tex2html_wrap_inline211 and a vector orthogonal to tex2html_wrap_inline211 .
    5.   Find an orthogonal basis for W.
    6. Find an orthogonal tex2html_wrap_inline233 matrix U, such that the corresponding linear transformation from tex2html_wrap_inline179 to tex2html_wrap_inline179 takes the tex2html_wrap_inline241 axis to the line spanned by tex2html_wrap_inline211 and the tex2html_wrap_inline241 , tex2html_wrap_inline247 coordinate plan to W. Hint: Use parts 5b and 5d.

  6. (16 points) Let W be the plane in tex2html_wrap_inline179 spanned by tex2html_wrap_inline255 and tex2html_wrap_inline257
    1. Find the projection of tex2html_wrap_inline259 to W.
    2. Find the distance from b to W.
    3. Find a least square solution of the equation Ax=b, where tex2html_wrap_inline269 is the tex2html_wrap_inline271 matrix with columns tex2html_wrap_inline273 and tex2html_wrap_inline275 . I.e., find a vector x in tex2html_wrap_inline279 which minimizes the length tex2html_wrap_inline281 .
    4. Find the coefficients tex2html_wrap_inline283 , tex2html_wrap_inline285 of the line tex2html_wrap_inline287 which best fits the three points tex2html_wrap_inline289 , tex2html_wrap_inline291 , tex2html_wrap_inline293 in the x,y plane. The line should minimize the sum tex2html_wrap_inline297 . Justify your answer!

  7. (16 points) The vectors tex2html_wrap_inline299 and tex2html_wrap_inline301 are eigenvectors of the matrix tex2html_wrap_inline303 .
    1. The eigenvalue of tex2html_wrap_inline211 is:

      The eigenvalue of tex2html_wrap_inline213 is:

    2. Find the coordinates of tex2html_wrap_inline309 in the basis tex2html_wrap_inline311 .
    3. Compute tex2html_wrap_inline313 .
    4. As n gets larger, the vector tex2html_wrap_inline317 approaches Justify your answer.

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Next: About this document

Eyal Markman
Mon May 21 09:20:58 EDT 2001