Sample Exam I

• Problem 1a. Let V be the vector space of real polynomials of degree less than or equal to 3.

  Consider the set E of elements in V given by E = {1 + x - x^2 + x^3, x + 2x^2 - x^3, 1 - x + x^2 + x^3, 4 - x - 2x^2 + 5x^3}.

  Is the set linearly dependent or independent. Show work and Explain.

• Problem 1b.

  Does E span V? Show work and Explain.

• Problem 2. Consider the 3 by 3 real matrix A with first row equal to (1 0 2),

  and second row equal to (2 1 1),

  and third row equal to (1 3 1),

  Determine 3 by 3 matrices L, D and U,

  with L lower triangular having 1's on the diagonal

  with U upper triangular having 1's on the diagonal

  and with D diagonal

  such that A = LDU.

• Problem 3a. Let A be a 4 by 4 symmetric matrix. Is A^2 also symmetric (explain)?

• Problem 3b.Making use of symmetry calculate the number of multiplications required to obtain A^2.

• Problem 4a.Solve the linear system Ax=b where A is the 3 by 4 matrix

  having first row (3 1 0 2)

  and second row (-1 1 2 1)

  and third row (0 -1 1 3)

  and b is the column vector (1 -1 2)

  Problem 4b. Determine the rank of the matrix A

  Answers and Hints for Sample Exam I

  • Problem 1a. Set a(1 + x - x^2 + x^3) +b( x + 2x^2 - x^3) +c( 1 - x + x^2 + x^3) + d( 4 - x - 2x^2 + 5x^3) = 0.

    This leads to a system of 4 equations in 4 unknowns. Determine solution space and check whether a non-trivial solution exists.

    System is a + c + 4d =0; a +b -c -d = 0; -a + 2b + c -2d =0 ; a -b +c +5d =0

    If so, dependent. Otherwise independent.

    Problem 1b. Solve the a + c + 4d = w_1; a +b -c -d = w_2 ; -a + 2b + c -2d = w_3 ; a -b +c +5d = w_4.

    If any row is produced with 0a + 0b + 0c + 0d then the linear expression(s) on the right side of any such row must be set to 0.

  • Problem 2. D = diagonal 3 by 3 with {1,1,8} on diagonal.

    U is 3 by 3 with rows (1, 0, 2), (0, 1, -3), (0, 0, 1).
    L is 3 by 3 with rows (1, 0, 0), (2, 1, 0), (1, 3, 1).

  • Problem 3a. (A^2)^T = (AA)^T = (A^T)(A^T) = (AA) (since A is symmetric). So A^2 is symmetric.

  • Problem 3b. I count 34 (but check this) as follows 10 terms which are squares (a_ij)^2 keeping in mind a_ij = a_ji by symmetry.

    And 12 products of form (diagonal)(off-diagonal)
    And 12 products of form (off-diagonal) (off-diagonal) which are not of previous types

  • Problem 4b. rank is easily checked to be 3.

  Sample Exam II

  • Problem 1a. Compute det A where A is the 4 by 4 matrix with entries:

    First row = (1, -1, 2, 1)

    Second row = (0, 1,1,2)

    Third row = (1, 1,1 0)

    Fourth row = (0, -1, 1, -1)

  • Problem 1b. Multiply the second row by 2 and the third column of A by -3 to obtain B where

    First row(B) = (1, -1, -6, 1)

    Second row = (0, 2, -6, 4)

    Third row = (1, 1, -3, 0)

    Fourth row = (0, -1, -3, -1)

    Determine det B^3.

  • Problem 2. Consider the linear map c from R^3 to R^3 which acts by taking the cross product with a = (1, -1, 3)

    More precisely, if v = (v_1, v_2, v_3) and w = (w_1, w_2, w_3) then v cross w = (v_2w_3-v_3w_2, v_3w_1-v_1w_3, v_1w_2-v_2w_1)

    And c(v) = v cross a

    Determine the matrix representing the map c with respect to the basis {(1,0,0), (1,1,0), (1,1,1)} of R^3.

  • Problem 3. Suppose the following are data points for (x,y):

    (-2,1), (-1,0), (0,3), (1,2), (2,2).

    Find the straight line y = mx + b that is the best least-suares fit for this data.

  • Problem 4. Use Cramer's rule to solve Ax=b.

    where A is the 3 by 3 matrix with first row (2, 1, 1),

    second row is (-1, 1, 1),

    and third row is (1, 1, -1),

    and b is the column vector in R^3 with entries (1, 0, 3).

  • Problem 5a. Let V be the inner product space of continuous functions on [-1, 1] with inner product

    defined by (f,g) = integral from -1 to 1 of f(t)g(t)dt.

    The functions a(x)=1 + x and b(x) = 1 - x^2 are linearly independent.

    Let W be the subspace of V spanned by {a, b}

    Use Gram-Schmidt to find an orthonormal basis {q^1(x), q^2(x)} for W.

  • Problem 5b. Determine a 2 by 2 upper triangular matrix R, such that

    (a(x), b(x)) = (q^1(x), q^2(x)) R

    Sample Final

    • Problem 1a. Let A be the matrix

      First row = (1, 1, 0, -1)

      Second row = (1, -1, 2, -1)

      Third row = (1, 1, 1, 1)

      Fourth row = (1, 1, -1, 0)

      Use Gaussian elimination to solve the linear system Ax = b

      where x = (x_1, x_2, x_3, x_4)^T and b = (1, -1, 0, 1)^T

    • Problem 1b. Find the L,D, U factorization of A

    • Problem 2. Let B be the matrix

      First row = (1, -1, 2, 1)

      Second row = (0, 1,1,2)

      Third row = (1, 1,1 0)

      Fourth row = (1, -2, 4, 4)

      Find a basis for the null space of B and

      the column space of B.

    • Problem 3. (PLEASE NOTE THERE IS AN ERROR IN THE ORIGINALSTATEMENT

      OF THIS PROBLEM.

      There should not be "square root" in the definition of the inner product below)

      Consider the inner product defined on

      the vector space V consisting of real polynomials

      of degree less than or equal to 2 given by

      (f(x), g(x)) = square root of the integral from 0 to 1

      of f(t)g(t)dt (CORRECTION: REMOVE "SQUARE ROOT")

      Start with the basis {1, 1+x, x+x^2} of V and use the Gram-Schmid process to find

      an orthonormal basis for V.

    • Problem 4. Use the method of least squares.

      Find the straight line y=mx + b (i.e., find m and b)

      which best fit the following 5 data points which have

      (x,y) coordinates as follows:

      (-2, -2), (-1,0), (0,1), (1,3), (2,4).

    • Problem 5.a Consider the quadratic form

      Q(x,y,z) = x^2 - 2xy + 4xz +3y^2 -6yz -2z^2

      Express Q as the difference of two sums of perfect sauares with positive coefficients.

    • Problem 5b. Use your answer in 5a to clasiffy the critical point

      of f(x,y,z) = 12 + x^2 - 2xy + 4xz +3y^2 -6yz -2z^2 at (0,0,0)

      as a local max, a local min, or a saddle point.

    • Problem 6.a. For the 3 by 3 matrix A given by

      First row = (5,2,2)

      Second row = (-3,0,-3)

      Third row = (1, 1, 4)

      Find a nonsingular matrix C such that D = C^{-1}AC is diagonal.

    • Problem 6.b. Use your answer in 6a to help find a full set of solutions

      to the first-order linear system of differential equations given by

      du/dx = Au, where u=(u_1(x), u_2(x), u_3(x))^T.

    • (DO OVER) Problem 6'.a. For the 3 by 3 matrix A' given by

      First row = (-1,-2,-2)

      Second row = (2,3,2)

      Third row = (-2,-2,-1)

      Find a nonsingular matrix C such that D = C^{-1}A'C is diagonal.

    • Problem 6'.b. Use your answer in 6a to help find a full set of solutions

      to the first-order linear system of differential equations given by

      du/dx = A'u, where u=(u_1(x), u_2(x), u_3(x))^T.

    • Problem 7a Diagonalize the 3 by 3 matrix R whose rows are given by

      First row = (1,0,0)

      Second row = (0,1,0)

      Third row = (-2,2,-1)

    • Problem 7b. What is R^{73}

    • Problem 7c. What is trace (R^{17})

    • Problem 8. For the 4 by 4 matrix B given by

      First row = (1, 1, 0, -1)

      Second row = (2, -1, 1, 0)

      Third row = (1, 1, 2, 1)

      Fourth row = (-2, 1, 0, 2)

      Determine det(B)

    • Problem 8b. Determine det(2B^{-1})