Homework Assignments for math 308A, summer 2011:

Homework assignments are due every Wednesday at the start of lecture.
Note: Homework problems in parentheses need not be turned in.

• Homework assignment 1 -- Due 6/29:
  • 1.1: 17, 20, 22, 29, 32
  • 1.2: 10, 16, 18, 21, 31, 35, 39, 41, 45, 49
  • 1.3: 4, 12, 18, 23, 24
• Homework assignment 2 -- Due 7/6:
  • 1.5: 4, 9, 13, 18, 21, 31, 45, (46), 52, 55, 57, 62, 68
  • 1.6: 7, 13, 17, 19, 26, 43
  • 1.7: 1, 2, 6, 12
• Homework assignment 3 -- Due 7/13:
  • 1.7: 24, 46, 47
  • 1.8: 2, 9, 27
  • 1.9: 2, 5, 9, 15, 39, 50, 51, 56*, 58, 69
    *To problem 56, add the following part (b). We will investigate how this matrix behaves in the case where n=2 using the following example. Let v= and find A. Draw a set of 2d coordinate axes, and graph the vector v and the line y=-x. Using the vectors , graph each of v, Av, w₁, Aw₁, w₂, Aw₂, w₃, Aw₃ on the same set of axes. Then describe in words what you think multiplication by A does to vectors.
    [Answer to 56(b): the effect of A is to reflect (flip) all vectors across the line y=-x, which is the line perpendicular to v. In R^3, the effect of such a matrix (from part a) is to reflect vectors across the plane perpendicular to v. Etc.]
• Homework assignment 4 -- Due 7/20:
  • 3.1: 13, 16, 25
  • 3.2: 1, 2, 3, (5), (8), (10), 11, 17, 18, 20, (27), (28), 29
    
  • 3.3: 3, 5, 6, 23, 33, 44, 50 [Hint for #44: there is no need to know specifically what the span is!]
  • 3.4: 29, 34, 36
• Homework assignment 5 -- Due 7/27:
  • 3.4: 1, (5), 9, 11
  • 3.5: 1, 4, 5, 11, (15), 19, 22, 25, 29, 33, 38, (39)
• Homework assignment 6 -- Due 8/3:
  • 3.6: 1, 5, 9, 13, A*, 28
    *Problem A: Consider the matrix . Find orthonormal bases for the kernel and range. (Notice: that says orthonormal, not merely orthogonal.)
  • 3.7: (1), 4, 5, 9, (13), 17, 18, 19, (26*), 30*, 32, (34), 46**
    [Hint for 19: [1 1]^T = e_1 + e_2, and T(e_1+e_2) = T(e_1)+T(e_2).]
    *The kernel and range of T are just the kernel and range of the representing matrix A. Same with rank and nullity (dim Ker A).
    **"Represent geometrically" just means draw them on a pair of coordinate axes.
• Homework assignment 7 -- Due 8/10:
  • 3.9: 11 - 14.
    [I would actually recommend finding orthonormal bases for the two subspaces ahead of time.]
  • 3.8: 1, 2, (7), 12, 17*
    *The point of this is that it shows least-squares polynomial fits to data are always unique.
  • 4.2: (1-4), (7), 8, (9), 10, 11, 17, 18, 19, 21
• Homework assignment 8 -- not collected:
  • 4.1: (17), 18*
  • 4.2: (24), 25, 26, 27, 28
  • 4.3: 1, (3), 7, 13, 14, (18), (21), (23), 28
  • 4.4: (1), 3, 5, 6, (8), (30)
  • 4.5: 3, 11, 14, 15, (22)
    *Notice: the rotation matrices have this form.
  • 4.7: 5, 11
  • 4.8: 3, 9*, (23), (24), (25)
    *Also find the eigenspace E₁.