3.4. Assignment #2: Using Vectors and Matrices (back to full lecture notes)

In this assignment you will solve problems involving vectors and matrices. Please submit a single file a2.* containing your solutions. The file extension may be anything you choose; please ask before submitting a file in an exotic or obscure file format (plain text is preferred).

  1. Consider the line L in R2 that passes through the following two vectors:
    u
    =
     
    9
    7
     
    v
    =
     
    1
    -5
     
    1. Using set notation, define L in terms of u and v.
    2. Determine whether the following vectors are on the line L:
       
      25
      31
       
      ,
       
      7
      −1
       
      ,
       
      −3
      −11
       
    3. Find all unit vectors orthogonal to L.
    4. Define the line L that passes through the origin and is orthogonal to L using an equation of the form:
      y
      =
      mx + b
      In other words, find m, b R such that:
      L
      =
      {
       
      x
      y
       
        |   y = mx + b }
  2. For each of the following collections of vectors, determine whether the collection is linearly independent. If it is, explain why; if not, show that one of the vectors is a linear combination of the other vectors in the collection.

    1. The following two vectors in R2:
       
      2
      1
       
      ,
       
      3
      2
       
    2. The following three vectors in R2:
       
      -2
      1
       
      ,
       
      1
      3
       
      ,
       
      2
      4
       
    3. The following three vectors in R4:
       
      2
      0
      4
      0
       
      ,
       
      6
      0
      4
      3
       
      ,
       
      1
      7
      4
      3
       
  3. Consider the following vectors in R3:
    u
    =
     
    3
    7
    9
     
    v
    =
     
    2
    −5
    3
     
    w
    =
     
    3
    4
    −3
     

    1. Compute the orthogonal projection of w onto the vector:
       
      1/√(12)
      1/√(12)
      1/√(12)
       
    2. Determine whether each of the following points lies on the plane perpendicular to u:
       
      3
      0
      -1
       
      ,
       
      7
      -9
      -5
       
      ,
       
      1
      1
      -1
       
    3. Extra credit: Given the vector v and w, let P be the plane orthogonal to v, and let Q be the plane orthogonal to w. Find a vector p R3 that lies on the line in R3 along which P and Q intersect, and provide a definition of the line.
  4. You decide to drive the 2800 miles from New York to Los Angeles in a hybrid vehicle. A hybrid vehicle has two modes: using only the electric motor and battery, it can travel 1 mile on 3 units of battery power; using only the internal combustion engine, it can travel 1 mile on 0.1 liters of gas (about 37 mpg) while also charging the battery with 1 unit of battery power. At the end of your trip, you have 1400 fewer units of battery power than you did when you began the trip. How much gasoline did you use (in liters)?

    You should define a system with the following dimensions:

    • net change in the total units of battery power;
    • total liters of gasoline used;
    • total number of miles travelled;
    • number of miles travelled using the electric motor and battery;
    • number of miles travelled using the engine.

    You should define a matrix M R3×2 to characterize this system. Then, write down an equation containing that matrix (and three variables in R), and solve it to obtain the quantity of gasoline.

    M
     
    x
    y
     
    =
     
    ?
    ?
    ?
     
  5. Suppose we create a very simple system for modelling how predators and prey interact in a closed environment. Our system has only two dimensions: the number of prey animals, and the number of predators. We want to model how the state of the system changes from one generation to the next.

    If there are x predators and y prey animals in a given generation, in the next generation the following will be the case:

    • all predators already in the system will stay in the system;
    • all prey animals already in the system will stay in the system;
    • for every prey animal, two new prey animals are introduced into the system;
    • for every predator, two prey animals are removed from the system;
    • we ignore any other factors that might affect the state (e.g., natural death or starvation).

    1. Specify explicitly a matrix T in R2×2 that takes a description of the system state in one generation and produces the state of the system during the next generation. Note: you may simply enter the matrix on its own line for this part of the problem, but you must also use it in the remaining three parts below.

    2. Show that the number of predators does not change from one generation to the next.

      T
       
      x
      y
       
      =
       
      x
      y'
       
    3. Determine what initial state [x0; y0] is such that there is no change in the state from one generation to another.

      T
       
      x0
      y0
       
      =
       
      x0
      y0
       

    4. Suppose that in the fourth generation of an instance of this system (that is, after the transformation has been applied three times), we have 2 predators and 56 prey animals. How many predators and prey animals were in the system in the first generation (before any transformations were applied)? Let [x; y] represent the state of the system in the first generation. Set up a matrix equation that involves [x; y] and matrix multiplication, and solve it to obtain the answer.