3.12. Assignment #3: Matrix Properties and Operations (back to full lecture notes)

In this assignment you will perform step-by-step algebraic manipulations involving vector and matrix properties and operations. You may not add new lines or premises above an "implies" logical operator. If you add any new premises, you will recieve no credit.

    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(uv) + v

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

      implies

      # ... put proof steps here ...

      (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
       
      M v
      =
       
      0
      0
      0
       
      M w
      =
       
      0
      0
      0
       
      v'
      =
      a u + b v + c w

      implies

      # ... put proof steps here ...

      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

      M u = v

      implies

      # ... put proof steps here ...

      # replace the right−hand side below
      # with an expression in terms of M and v
      u = ?

    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

      # ... put proof steps here ...

      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
      =
      ?
      B
      =
      ?
      C
      =
      ?
      D
      =
      ?

      implies

      # ... put proof steps here ...

      A
       
      ab
      cd
       
      =
       
      a+c b+d
      c d
       
      B
       
      ab
      cd
       
      =
       
      a b
      c+a d+b
       
      C
       
      ab
      cd
       
      =
       
      ta tb
      c d
       
      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
      =
      ?
      A
      =
      ?
      B
      =
      ?
      C
      =
      ?
      D
      =
      ?
      E
      =
      ?

      implies

      # ... put proof steps here ...

      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
      A
      =
      ?
      B
      =
      ?
      C
      =
      ?
      D
      =
      ?

      implies

      # ... put proof steps here ...

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