• Read sections 1.1-1.4 in the text. (My Notes on Row Reduction cover the essential material in 1.1 and 1.2.)
• Do the following problems.
  1. Use row reduction methods to solve the system S below. Put the solution in parametric form. Identify the leading and non leading variables. Find the solution to the corresponding homogeneous system. Again, put the solution in parametric form.

  2. Find the reduced row echelon form for the matrix B.

  3. For the matrices R, S, u, v, find the combinations below, if possible; if not, state why you can't.

     (a) 6R+3S

     (b) RT

     (c) RS

     (d) SR

     (e) vu

     (f) uv

     (g) Ru

     

Assignment 2 - Due Friday, June 7, 2013.

• Read sections 1.3, 1.4, and 1.5 (You can skip the applications for now. We'll do some, but later on.)
• Do the following problems.

  1. §1.3 (pp. 43): 10(a,c). To do these, first use the basic matrix trick (boxed eqn. on p. 34) to find Ax for all x. Then, use Theorem 3.1 to answer the questions.

  2. §1.3 (pp. 43-44): 13(a,b). To do part (b), first be aware that the answer is not just the last column in the matrix displayed in part (a). (Why?) Second, use the basic matrix trick to write b as a linear combination of the columns of A. You'll solve the problem if you can find a solution x that is nonzero in only the first and third entries.

  3. §1.4 (p. 56): 9

  4. §1.4 (p. 56): 12 (Use row reduction on [A|I]. You may assume a₁₁ ≠ 0.)

  5. §1.4 (p. 56): 13(b)

  6. §1.5 (p. 66-67): 10(f,h)

Assignment 3 - Due Tuesday, June 11, 2013.

• Read sections 2.1, 2.2, 3.1, 3.2
• Do the following problems.
  1. §1.5 (p. 66): 5(a,b,c) (In (a) and (b), compare the matrices involved to see what row operation was used to go from A to B or C, then write down the elementary matrix corresponding to the row operation.)

  2. §1.5 (p. 66): 7(a,b)

  3. Suppose that A is a singular 3×3 matrix. Show that there is always a column vector x ≠ 0 such that Ax = 0 . (Hint: if A is singular, what does its reduced echelon form look like?)

  4. §2.1 (p. 94): 3(d,g) (Use the cofactor method for these.)

  5. §2.1 (p. 95): 5 (Use the cofactor method for these.)

  6. §2.2 (p. 101): 3(e,f) (Use row reduction methods for these.)

  7. §2.2 (p. 101): 4

Assignment 4 - Due Thursday, June 13, 2013.

• Read sections 3.3, 3.4.
• Do the following problems.
  1. §3.2 (p. 131): 1(d)

  2. §3.2 (p. 131): 2(a,b)

  3. §3.2 (p. 131): 3(d,f)

  4. §3.2 (p. 131): 4(b)

  5. §3.2 (p. 131): 6(d,e)

  6. §3.2 (p. 132): 12(d,e)

  7. §3.2 (p. 132): 13(a,b)

  8. §3.2 (p. 132): 16(a,b)

Assignment 5 - Due Tuesday, June 18, 2013.

• Read sections 3.6, 4.1, the notes on Coordinate Vectors, and the notes on Methods for Finding Bases.
• Do the following problems.
  1. §3.3 (p. 143): 2(c,e)

  2. §3.3 (p. 143): 4(c)

  3. Use coordinate vectors to determine whether $S=\{x^3+1,x+x^2,x^3-1, 1+x+x^2+x^3\} \subset \mathcal P_4$ is linearly dependent or linearly independent.

  4. §3.4 (p. 149): 2(c,e)

  5. §3.4 (p. 149): 14(b,d)

  6. §3.6 (p. 165): 1(b)

  7. §3.6 (p. 166): 9

  8. §3.6 (p. 166): 15(a,b)

Assignment 6 - Due Tuesday, June 25, 2013.

• Read sections 6.1, 6.3, 6.4, 5.4, and the notes on Change of Basis
• Do the following problems.
  1. §4.1 (p. 182): 4
  2. §4.1 (p. 182): 6(b,d)
  3. §4.1 (p. 183): 19(a,b,c)
  4. §4.2 (p. 195): 5(a)
  5. §4.2 (p. 196): 15
  6. Let L : P₃ → P₃ be defined by L[p]= (2x² + x + 1)p'' − (3x − 1)p' + 2p.
     1. Show that L is linear.
     2. Find the matrix A of L relative to the standard basis E = {1, x, x²}.
     3. Find bases for the null space, and column space of A.
     4. Use the bases you found above to write down bases for the null space and image of L.
     5. Use the matrix A to solve for the polynomial p if L[p] = 3 − x.
  7. Find the transition matrix S_{E → F} if E = {1, x, x²} and F = {1-x, 2x+x², x²-1}
  8. In R², find the transition matrix S_{E → F} if E = {(1 -1)^T, (1 1)^T} and F = {(2 -1)^T, (1 2)^T}.

Assignment 7 - Due Friday, June 28, 2013.

• Read sections 8.4, 10.1, and 10.2.
• Do the following problems.
  1. §6.1 (p. 308): 1(c,d,h)
  2. §6.1 (p. 308): 6
  3. §6.1 (p. 308): 13
  4. §6.1 (p. 309): 16 (See pgs. 306 and 307.)
  5. §6.3 (p. 336): 1(a,d,e)
  6. §6.3 (p. 336): 2(d)
  7. §6.3 (p. 336): 7
  8. Let L : P₃ → P₃ be defined by L[p]= (1-x² )p'' − 2xp'. We want to find the eigenfunctions and eigenvalues for L.
     1. Show that L is linear.
     2. Find the matrix A of L relative to the standard basis E = {1, x, x²}.
     3. Find the eigenvalues and eigenvectors for A.
     4. Use the previous part to write down the eigenvalues and corresponding "eigenfunctions" for L. In one sentence, explain why you can do this. (The eigenfunctions that you've found are called Legendre polynomials. These are important in physics, geophysics, engineering, and statistics.)

Assignment 8 - Due Tuesday, July 2, 2013.

• Read sections 11.1, 11.2, 11.3. In section 11.1, you need to know the following:
  • Parametrizations for a spheres, cylinders, planes, cones, amd surfaces given by $z=f(x,y)$.
  • The standard normal for a parametrized surface: $\mathbf N = \frac{\partial \mathbf X}{\partial s}\times \frac{\partial \mathbf X}{\partial t}$. (Colley uses $\mathbf T_1 = \frac{\partial \mathbf X}{\partial s}$ and $\mathbf T_2 = \frac{\partial \mathbf X}{\partial t}$.)
  • Tangent plane: This is the plane through the point $\mathbf X(s,t)$ and perpendicular to the standard normal $\mathbf N(s,t)$.
  • The unit normal: $\mathbf n = \mathbf N/|\mathbf N|$
  • Area element: $dS = |\mathbf N|ds dt$.
  • Vector area: $d\mathbf S = \mathbf N dsdt$

• Do the following problems.
  1. §10.1 (p. 675): 24
  2. §10.1 (p. 675): 29
  3. §10.2 (p. 685): 10
  4. §10.2 (p. 684): 22(a,c)
  5. §11.2 (p. 739): 14
  6. §11.2 (p. 739): 22
  7. §11.3 (p. 755): 4
  8. §11.3 (p. 755): 5
  9. §11.3 (p. 755): 8
  10. §11.3 (p. 755): 11

Updated 6/27/2013 (fjn).