MA-305 Homework 4

Due at 2:35 pm, Tuesday, March 24, 1998


Calculations necessary for these problems may be done either by hand or with Maple. Solutions may be submitted in person in class, or you may email an ASCII text, Maple Text, or Maple Worksheet (.mws) to the TA, John Haws (jchaws@eos.ncsu.edu).

Remember: If you have a question, you may find the answer in the Forum.

  1. Define the set P3 by:
    P3 = {a + bx + cx2 + dx3 | a, b, c, d are Rational Numbers}.
    1. Verify that P3 is a vector space with normal polynomial addition and scalar multiplication over polynomials.

    2. Verify that (-1)V = - V, for all V in P3.

    Note: We introduced this problem in Lecture 14, Slide 1.

  2. *Determine if the given set V is a vector space. Justify your answer.

    1. The set V of all pairs of real numbers, i.e.
      V = { (x1,x2) | x1,x2 are Real Numbers}.
      with usual scalar multiplication, and with addition defined by
      (x1,x2) + (y1,y2) = (x1 + y1 + 1, x2 + y2 + 1)

    2. The set V of all ordered triples {(x1,x2,x3)} with usual addition, and with scalar multiplication defined by
      r(x1,x2,x3) = (2rx1,2rx2,2rx3)
      for all scalars r.

  3. *A Vector Space V and a subset S are given in the exercises below. For each, determine whether S is a subspace of V. Justify your answer.

    1. Let V = P3 from Ex. 1 above. Define S by
      S = { ax + bx3 | a and b are Rational Numbers}.

    2. Let V = M22 be the vector space of all 2x2 matrices with real number entries, under the usual operations of matrix addition and scalar multiplication. Define S by
      S = { [
      [       		0
      a       		b
      0       		]
      ]       		| a and b are real numbers }

  4. *Find a set of vectors spanning the solution space of Ax=0, where
    A = [
    [
    [     		2
    4
    14     		7
    -3
    15     		-1
    4
    7     		]
    ]
    ]

  5. *Determine which of the following sets of vectors are linearly independent. Justify your answer. For those sets that are linearly dependent, express "extra" vectors in terms of the others.

    1. In R3, {u, v, w} = {(2, -1, 3), (3, 4, 1), (2, -3, 4)}

    2. In R4, {u, v, w} = {(2, 3, -1, 2), (-6, 1, 4, 5), (12, 8, -7, 1)}
    3. In R4, {v1, v2, v3, v4} = {(1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1), (1, 0, 0, 1)}

  6. *Determine which of the following sets of vectors form a basis for the given vector space. Justify your answer.

    1. For R3, v1 = (3, -2, 1), v2 = (2, 3, 1), v3 = (2, 1, -3)
    2. For R2, v1 = (2, 0), v2 = (3,3)
    3. For P3 (defined in Exercise 1 above),   p1 = 1 + x,   p2 = 1 - x2,   p3 = x3,      p4 = 1 - x

  7. **Find a basis for the Null Space associated with the following matrix A:
    A = [
    [
    [     		1
    3
    2     		2
    6
    4     		0
    1
    1     		2
    9
    7     		1
    6
    5     		]
    ]
    ]

    Bonus: How many calls to the recursive determinant function defined in Slide 3, Lecture 13 will be made if we use memoization?

*: problem from Introductory Linear Algebra with Applications, by Kolman and Hill

**: problem from Matrix Analysis and Linear Algebra, by Carl Meyer