Math 311 (Honors) Assignments - Spring 2006

Assignment 1 - Due Friday, 27 January.

• Read Notes on Row Reduction and The Rank of a Matrix.
• Do the following problems. The matrices you need are listed at the end of this assignment.
  1. 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

  2. Find a 2×2 matrix C such that C_jk ≠ 0 and C² = 0_2×2.

  3. Find the reduced echelon form for the matrix B. In addition, find rank(B) and find the leading columns of B.

  4. Let M be an arbitrary m×n matrix and let e_j be the n×1 column vector whose entries are 0, except for the j^th, which is 1. Use the definition of matrix multiplication to show that Me_j is the j^th column of M.

  5. Show that row rearrangement R_i <-> R_j can be achieved by operations involving elementary multiplicaion and elementary modification. (Hint: it can be done in four operations.)

  6. 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.
  

Assignment 2 - Due Wednesday, 8 February.

• Read Notes on Row Reduction and The Rank of a Matrix.
• Do the following problems. The matrices you need are listed at the end of this assignment.
  1. The matrix [A|b] below is the augmented matrix for a system of linear equations. Find the reduced echelon form of [A|b], then find rank(A) and rank([A|b]). Does A have full rank? Is the system consistent? If so, does the system have a unique solution or are there many solutions? State the leading columns of A. Solve the corresponding homogeneous system; put the solution in parametric form.

  2. Repeat the previous question using the complex matrix [C|d] in place of [A|b].

  3. Determine whether the vectors in the set S₁ are LI or LD. If they are LD, find a nontrivial linear combination of them that vanishes.

  4. A communications company does an analysis of discrete signals of length five. They find that nearly all signals they encounter can be represented as a linear combination of the three column vectors in the set S₂. This allows them to transmit three numbers rather than five, except for the occasional signal that can't be represented this way.

     1. Determine whether the discrete signal s can be represented as a linear combination of the three vectors. If so, find three numbers that represent this signal.

     2. When a signal can be represented in this way, are the three numbers unique? Does it matter? Explain.

     3. Use a stem plot or bar graph to plot the signal and three column vectors.

  5. Show that the three 4×1 column vectors in S₃ are linearly independent. Show that it is impossible to find two vectors u and v such that each of the three vectors in S₃ is a linear combination of u and v.

  6. For each of the matrices P and Q, either find the inverse or show that it doesn't exist.
  

Assignment 3 - Friday, 17 February.

• Read chapter 1 and sections 2.1-2.4 in the text.
• Do the following problems. The matrices you need are listed at the end of this assignment.
  1. For the matrices A and B below, find det(A) and det(B) using the method employing row operations. For each matrix, answer the following questions.
     1. Is the matrix invertible?
     2. Are the columns of the matrix LI or LD?
     3. Are the rows of the matrix LI or LD?
     4. Is there a nonzero vector x such that the matrix times the vector is 0?
     5. Is the rank of the matrix equal to 4?

     

  2. For the matrix B from the previous problem, use the row expansion method to find det(B).

  3. Use Cramer's rule to find x₃, given that Bx = (0 1 2 1)^T. You may use det(B) from above.

  4. For the matrices below, verify that det(PQ) = det(P)det(Q). Use any method to find the determinants involved.

     

  5. Let L be a lower triangular n×n matrix. (Thus, L_j,k = 0 for k > j.) Use the permutation definition of determinant to show that det(L) = L_1,1 L_2,2 ... L_n,n

  6. Chapter 1, problem 4, pg. 8.

  7. Section 2.1, problem 5, pg. 13.

Assignment 4 - Monday, 6 March.

• Read sections 3.1-3.4 in the text. (We are temporarily skipping 2.5.)
• Do the following problems.
  1. Section 2.1, problems 8, 9, pg. 13.

  2. Section 2.2, problems 5, 8, 12, pgs. 21-22.

  3. Section 2.3, problems 6, 9, 10, pg. 25.

  4. Section 2.4, problems 7, 9, 10, pg. 29.

  5. Bonus. The second order linear differential equation y′′ + p(t)y′ + q(t)y = f(t), a ≤ t ≤ b, where the coefficients p and q are continuously differentiable. Let Y = (y y′ )^T and F = (0 f )^T .
     1. Show that Y′ = AY + F, where

        

     2. Let u and v be linearly independent solutions to this ODE when f=0. (This is the homogeneous case.) Also, let U = (u u′ )^T and V = (v v′ )^T . Show that for each t in [a,b] the set B := {U, V} is a basis for R². (Hint: look at the Wronskian W=W(u,v). How is it related to U and V being LI? Can it vanish?)

     3. Show that the coordinate vector Z = [Y]_B =: (z₁ z₂)^T satisfies the equation [U V]Z′ = F. Solve this equation using Cramer's rule. Compare the answer with the usual formulas arising in the variation of parameters technique for solving non homogeneous linear ODEs.

Assignment 5 - Wednesday,22 March.

• Read sections 3.5, 4.1-4.3 in the text.
• Do the following problems.
  1. Section 3.1, problems 3, 5, 8, 16, 17, pgs. 41-42.

  2. Section 3.2, problems 3, 4, 8, 12, pgs. 45-46.

  3. Section 3.3, problems 3, 4, 7, pg. 49.

Assignment 6 - Monday, April 2.

• Read sections 4.5, 4.6 4.8 in the text.
• Do the following problems.
  1. Section 3.5, problems 7, 8, 9, pgs. 55-56.

  2. Section 4.1, problems 3, 6, pg. 59.

  3. Section 4.2, problem 1, pg. 61.

  4. Section 4.3, problems 3, 8, 11, pg. 67.

  5. Consider the linear differential operator L(u) = u′′ − 2tu′.
     1. Show that L: C₃[t] → C₃[t]. (Here C₃[t] is the set of cubic polynomials with complex coefficients.)

     2. Find A, the matrix of L with respect to the standard basis E = {1, t, t², t³}.

     3. Find the eigenvalues and eigenvectors of A. Use these to find the same things for L. (The polynomials that you get are called Hermite polynomials.)

     4. Directly find the matrix of L relative to a basis from composed of the eigenvectors (or, eigenfunctions) that you found above.

Assignment 7 - Monday, April 9.

• Read sections 5.3, 5.7 (nonlinear forces) and 6.1 in the text.
• Do the following problems.
  1. Section 4.3, problem 14, pg. 67.

  2. Section 4.5, problems 1, 7, 9, pg. 76.

  3. Section 4.6, problems 3, 6, pg. 82.
  4. Section 5.3, problems 6, 7 (bonus), 8, pgs. 116.

Assignment 8 - Wednesday, April 26.

• Read sections 6.2-6.5, 6.9 in the text.
• Do the following problems.
  1. Section 5.7, problems 6, 7, pg. 142. (Bonus)
  2. Section 6.1, problem 6 (also calculate the angle between them), pg. 149.

  3. Section 6.2, problems 4, 7, pg. 152.

  4. Section 6.3, problem 6, pg. 158.

  5. Section 6.4, problems 2, 3, pg. 163.

  6. Section 6.5, problems 9, 10, pg. 168-169.

Updated 4/20/06 (fjn).