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Math 235 Solution of 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_inline494 tex2html_wrap_inline496

    a) The rank of A is 3.

    b) A basis for the null space of A: The reduced echelon form is tex2html_wrap_inline502 The free variables are tex2html_wrap_inline504 and tex2html_wrap_inline506 . The general solution is

    displaymath470

    and the two column vectors on the right hand side are a basis for Null(A).

    c) A basis for the column space of A: The first, third and fifth columns of A are its pivot columns. They are a basis for the column space of A.

    d) A basis for the row space of A: Take the three non-zero rows of B.

  2. (4 points) The null space of the tex2html_wrap_inline520 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!

    a) tex2html_wrap_inline528 . Thus, tex2html_wrap_inline530 .

    b) The dimension of the column space is equal to the dimension of the row space, which is 4.

  3. (15 points)
    1. Use the cofactor expansion along the third column (which has two zero entries) to calculate the characteristic polynomial of the matrix tex2html_wrap_inline532

      tex2html_wrap_inline534

    2. A basis of tex2html_wrap_inline536 consisting of eigenvectors of A:

      The three eigenvalues are -1, 1 and 2. The -1-eigenspace is Null(A+I) and it is spanned by tex2html_wrap_inline550

      The 1-eigenspace is Null(A-I) and it is spanned by tex2html_wrap_inline556

      The 2-eigenspace is Null(A-2I) and it is spanned by tex2html_wrap_inline562

      Hence, tex2html_wrap_inline564 is a besis of eigenvectors of A.

    3. The tex2html_wrap_inline568 matrix tex2html_wrap_inline570 , whose columns are the three eigenvectors above, satisfies

      displaymath471

  4. (12 points)
    1. tex2html_wrap_inline572 is diagonalizable because it is a symmetric matrix with real entries.
    2. The matrix tex2html_wrap_inline574 has only one eigenvalue 2. The 2-eigenspace tex2html_wrap_inline578 is one-dimensional. Hence, the matrix A does not have two linearly independent eigenvectors. It is thus not diagonalizable.
    3. The characteristic polynomial of tex2html_wrap_inline582 is tex2html_wrap_inline584 Hence, A does not have any real eigenvalue. In particular, A is not diagonalizable via matrices with real entries.

  5. (22 points) Let W be the plane in tex2html_wrap_inline536 spanned by tex2html_wrap_inline594 and tex2html_wrap_inline596
    1.   The length of tex2html_wrap_inline598 is tex2html_wrap_inline600
    2.   The distance between the two points tex2html_wrap_inline598 and tex2html_wrap_inline604 in tex2html_wrap_inline536 is: tex2html_wrap_inline608
    3.   The subspace tex2html_wrap_inline610 orthogonal to W is:

      tex2html_wrap_inline614 It is spanned by tex2html_wrap_inline616 Hence, tex2html_wrap_inline618 is a unit vector in tex2html_wrap_inline610 .

    4.   The projection of tex2html_wrap_inline604 to the line spanned by tex2html_wrap_inline598 is:

      displaymath472

    5.   Write tex2html_wrap_inline604 as the sum of a vector parallel to tex2html_wrap_inline598 and a vector orthogonal to tex2html_wrap_inline598 :

      Subtract from tex2html_wrap_inline604 its orthogonal projection to the line spanned by tex2html_wrap_inline598 . The resulting vector

      displaymath473

      will be perpendicular to the line spanned by tex2html_wrap_inline598 . Thus,

      displaymath474

      is the required decomposition of tex2html_wrap_inline604 .

    6.   An orthogonal basis for W is provided by tex2html_wrap_inline642
  6. (16 points) Let W be the plane in tex2html_wrap_inline536 spanned by tex2html_wrap_inline648 and tex2html_wrap_inline650
    1.   The vectors tex2html_wrap_inline652 and tex2html_wrap_inline654 provide an orthogonal basis for W. Hence, the projection of tex2html_wrap_inline658 to W is:

      displaymath475

    2. The distance from b to W is

      displaymath476

    3.   A least square solution to the equation Ax=b, where A is the tex2html_wrap_inline670 matrix with columns tex2html_wrap_inline652 and tex2html_wrap_inline654 , is a vector x in tex2html_wrap_inline678 which minimizes the length tex2html_wrap_inline680 . We can calculate it in two ways:

      First Method: (using part 6a). The vector Ax will be in the subspace W. The point in W closest to b is tex2html_wrap_inline690 . Solve tex2html_wrap_inline692 . Since tex2html_wrap_inline694 , then tex2html_wrap_inline696 .

      Second Method: We can calculate x directly as the solution of the equation

      eqnarray254

    4. Find the coefficients tex2html_wrap_inline700 , tex2html_wrap_inline702 of the line tex2html_wrap_inline704 which best fits the three points tex2html_wrap_inline706 , tex2html_wrap_inline708 , tex2html_wrap_inline710 in the x,y plane. The line should minimize the sum tex2html_wrap_inline714 .

      The value tex2html_wrap_inline716 can be written as the dot product of tex2html_wrap_inline718 with tex2html_wrap_inline720 . Hence, the sum tex2html_wrap_inline714 can be written as the square of the norm of the vector

      displaymath477

      Plugging in the coordinates tex2html_wrap_inline724 of the three points given, we get

      displaymath478

      The problem reduces to part 6c with tex2html_wrap_inline726

  7. (16 points) The vectors tex2html_wrap_inline728 and tex2html_wrap_inline730 are eigenvectors of the matrix tex2html_wrap_inline732 .
    1. We calculate tex2html_wrap_inline734 . Hence, the eigenvalue of tex2html_wrap_inline598 is 1.

      The eigenvalue of tex2html_wrap_inline604 is .3 because tex2html_wrap_inline744

    2. The coordinates tex2html_wrap_inline702 , tex2html_wrap_inline748 of tex2html_wrap_inline750 in the basis tex2html_wrap_inline752 are its coefficients as a linear combination tex2html_wrap_inline754 We find them by row reduction

      displaymath479

      So, tex2html_wrap_inline756 and tex2html_wrap_inline758 .

    3. tex2html_wrap_inline760 .
    4. tex2html_wrap_inline762 As n gets larger, the vector tex2html_wrap_inline766 approaches the zero vector. Hence, as n gets larger, the vector tex2html_wrap_inline770 approaches tex2html_wrap_inline772 which is equal to tex2html_wrap_inline774 .




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Eyal Markman
Mon Dec 11 07:53:41 EST 2000