Assignment #2: Matrix Algebra (back to full lecture notes)

In this assignment you will perform step-by-step algebraic manipulations involving vector and matrix properties and operations. For the word problems, you must solve them by setting up a system with dimensions, introducing a matrix characterizing the relationships between the dimensions in that system, and then solving for the system states that contain the information being sought in the problem statements.

    1. Finish the argument below that shows that matrix multiplication of vectors in R3 preserves lines.

      M R3×3, ∀ u,v,w R3, ∀ s R,
      w = s(u-v) + v   and

      (w) is on the line defined by (u) and (v)

      implies

      # ...

      (M w) is on the line defined by (M u) and (M v)

    2. Finish the argument below that shows that if multiplication by a matrix M maps three vectors to [0; 0; 0], it maps any linear combination v' of those vectors to [0; 0; 0].

      M R3×3, ∀ u,v,w,v' R3, ∀ a,b,c R,
      M u
      =
       
      0
      0
      0
       
        and
      M v
      =
       
      0
      0
      0
       
        and
      M w
      =
       
      0
      0
      0
       
        and
      v'
      =
      a u + b v + c w

      implies

      # ...

      M v' =
       
      0
      0
      0
       


      Extra credit: if u, v, and w are setwise linearly independent, what can you say about the matrix M? Be as specific as possible.

    3. Show that if a matrix M in R2×2 is invertible, there is a unique solution for u given any equation M u = v with v R2.

      M R2×2, ∀ u,v R2,
      (M) is invertible   and

      M u = v

      implies

      # ...

      # replace the right-hand side below
      # with an expression in terms of M and v
      u = \undefined

    4. Show that if the column vectors of a matrix in R2×2 are linearly dependent, then its determinant is 0.

      a,b,c,d R,
      (
       
      a
      c
       
      ) and (
       
      b
      d
       
      ) are linearly dependent

      implies

      # ...
      det 
       
      ab
      cd
       
      = 0

      Extra credit: you may include the opposite direction of this proof for extra credit (build a new, separate argument in which det  [a,c; c,d] = 0 is found above the implies and `([a;c]) and ([b;d]) are linearly dependent` is at the end of the argument below it).

  1. In this problem, you will define explicit matrices that correspond to elementary row operations on matrices in R2×2.
    1. Find appropriate matrices A, B, C, and D in R2×2 to finish the following argument.

      A,B,C,D R2×2, ∀ a,b,c,d,t R,
      A
      =
      \undefined   and
      B
      =
      \undefined   and
      C
      =
      \undefined   and
      D
      =
      \undefined

      implies

      # ...
      A
       
      ab
      cd
       
      =
       
      a+c b+d
      c d
       
        and
      B
       
      ab
      cd
       
      =
       
      a b
      c+a d+b
       
        and
      C
       
      ab
      cd
       
      =
       
      ta tb
      c d
       
        and
      D
       
      ab
      cd
       
      =
       
      a b
      tc td
       


    2. Use the matrices A, B, C, and D from part (a) with matrix multiplication to construct a matrix E that can be shown to satisfy the last line in the following argument.

      A,B,C,D,E R2×2, ∀ a,b,c,d,t R,
      t
      =
      \undefined   and
      A
      =
      \undefined   and
      B
      =
      \undefined   and
      C
      =
      \undefined   and
      D
      =
      \undefined   and
      E
      =
      \undefined

      implies

      # ...
      E
       
      ab
      cd
       
      =
       
      cd
      ab
       


    3. Extra credit: the row operations defined by A, B, C, and D are all invertible as long as t ≠ 0; prove this for t = -1 by showing that the matrices A, B, C, and D are invertible.

      A,B,C,D R2×2, ∀ t R,
      t
      =
      -1   and
      A
      =
      \undefined   and
      B
      =
      \undefined   and
      C
      =
      \undefined   and
      D
      =
      \undefined

      implies

      # ...

      (A) is invertible   and
      (B) is invertible   and
      (C) is invertible   and
      (D) is invertible

  2. 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 in 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.

    x,y,z R,
     
    \undefined \undefined
    \undefined \undefined
    \undefined \undefined
     
     
    x
    y
     
    =
     
    \undefined
    \undefined
    \undefined
     


    implies

    # ...

    z = \undefined

  3. 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 R2×2, ∀ x,y,y' R,
      # replace \undefined with explicit matrix from part (a)
      T = \undefined

      implies

      # ...

      # In the conclusion below, you may replace y'
      # with the correct expression you may NOT replace
      # x.
      T
       
      x
      y
       
      =
       
      x
      y'
       


    3. Determine what initial state [x; y] is such that there is no change in the state from one generation to another.

      T R2×2, ∀ x,y R,
      # replace \undefined with explicit matrix from part (a)
      T
      =
      \undefined   and
       
      x
      y
       
      =
      \undefined # find the explicit vector

      implies

      # ...

      T
       
      x
      y
       
      =
       
      x
      y
       


    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.