[Solved] MATH 141: Linear Analysis I Homework 11

1. (if not done last week) Calculate the following determinant using Gaussian elimination. To make calculation by hand easier, you may use any row operations that we may or may not use in our standard forward Gaussian elimination process.

2. (Strang 4th ed. 4.3 #5) Let F_n be the determinant of the 1,1,1 tridiagonal matrix (n-by-n):

(b) By expanding in cofactors along row 1, show that F_n = F_n1 + F_n2.

This yields the Fibonacci sequence 1,2,3,5,8,13, for the determinants.

3. (Strang 4th ed. 4.3 #6) Suppose A_n is the n-by-n tridiagonal matrix with 1s on the three diagonals:

Let D_n be the determinant of A_n; we want to find it.

• Expand in cofactors along the first row to show that D_n = D_n1 D_n2.
• Starting from D₁ = 1 and D₂ = 0, find D₃,D₄,,D₈. By noticing how these numbers cycle around (with what period?) find D₁₀₀₀.

4. Earlier in the semester, we discussed how we could think of the modes of transportation in Finding Gauss problem as a basis for R². The travel of the hover board was given by and that of the magic carpet by . We found, for instance, that Gausss cabin, located 107 miles East and 64 miles North of your home, could be reached by riding the hover board forward for 30 hours and the magic carpet forward for 17 hours. Thus, Gausss location ~x could be described in two ways: using the standard basis, call it , Gauss lives at. Using the modes of transportation as a basis, call it, Gauss lives

at.

• Suppose Uncle Cramers house w~ is located at. Describe this location as a vector in the travel coordinate system . Explain your work.
• Suppose you visit a museum ~v located at. Describe the location of the museum as a vector

in the standard coordinate system . Explain your work.

• Express each of the following in terms of the travel basis and.
• Express each of the following in terms of the standard basisand.

5. Consider the following plane with both the red and black coordinate systems. The red axes correspond to what are normally thought of as the linesand y = 4x. Let the two red vectors highlighted along the red axis be the basis vectors for a new red basis, called.
   • Write the coordinates of each of the points (a)-(f) relative to both the red basis red and the standard black basis black.
   • Determine a matrix that will
     1. Rename points from the red basis as points in the black one (call it P).
     2. Rename points from the black basis as points in the red one (call it Q).

How are P and Q related?

• Consider now the linear transformation S : R² R² that has a stretch factor of 2 corresponding to the lineand a stretch factor of 1 corresponding to the line y = 4x.
  1. Determine what happens to the vector under this transformation. Represent the answer using coordinates in the black basis. Explain all steps in your solution approach. ii. Determine what happens to the vector under this transformation. Represent the answer using coordinates in the red basis. Explain all steps in your solution approach.
• Consider the following diagram, where ~y is the image of ~x under the transformation S, the matrices B and D are the stretch matrices for S in black and red respectively, and the matrix P renames vectors from red coordinates to black coordinates:
  1. Find the matrices B, D, and P that correspond to this diagram for this particular problem. For each of these, give at least one sentence explaining how you found them. How are the matrices B, D, and P related to each other?
  2. Pick one of the questions in part (c) above, either (c)i or (c)ii, and explain, using the above diagram, how there are two different methods in which you could have found the answer to that question.

6. We use the same red and black coordinate systems as in the previous problem, and S : R² R² denotes the same linear transformation that has a stretch factor of 2 corresponding to the lineand a stretch factor of 1 corresponding to the line y = 4x.
   • Without trying to find any matrices, describe in words, the stretch factors and stretch directions for the linear transformation S². Recall that S² = S S, or equivalently, S²(~x) = S(S(~x)) for any ~x in the domain of S.
   • Consider a diagram similar to that in problem 5(d) but for S², where ~z is the image of ~x under the transformation S², the matrices B_S2 and D_S2 are the stretch matrices for S² in black and red respectively.

Find the matrices B_S2 and D_S2, using the same method with which you found matrices B and D for

S.

• How is the matrix B_S2 related to B in problem 1? What about D_S2 to D? How are B_S2, D_S2, and P related?
• If we ask the same questions as those in parts (a), (b), and (c) about S^k for some integer k > 2, what do you think the answers are?

7. (optional) We consider three linear transformations R² R² and their matrices in this problem.
   • (from lecture) T : R² R² has a stretch factor of 2 corresponding to the line y = 3x and a stretch factor of 1 corresponding to the line y = x.
   • (from problem 5) S : R² R² has a stretch factor of 2 corresponding to the lineand a stretch factor of 1 corresponding to the line y = 4x.
   • (a new one) R : R² R² has a stretch factor of 1/4 corresponding to the line y = 3x and a stretch factor of 2 corresponding to the line y = x.

The following diagrams show the naming convention for various matrices associated to T, S, and R.

for

for

• (We have found all matrices in the first two diagrams in lecture or in problem 1.) Find matrices C, D_R, and P_R in the last diagrm for the linear transform R.
• Calculate two matrix products AB and BA to verify that AB 6= BA. (We say that A and B do not commute.)
• Calculate two other matrix products AC and CA to verify that AC = CA. (We say that A and C commute.)
• Study the above descriptions of T, S, R and the associated diagrams, to construct an explanation on why A and B do not commute while A and C