Assignment #1: Vector Algebra (back to full lecture notes)

In this assignment you will perform step-by-step algebraic manipulations involving vectors and vector operations. In doing so, you will assemble machine-verifiable proofs of algebraic facts. Please submit your solutions by email in a single text file.

Working with machine verification can be frustrating; minute operations that are normally implied must often be explicit. However, the process should familiarize you with common practices of rigorous formal reasoning in mathematics: finding, applying, or expanding definitions to reach a particular formula that represents a desired argument or solution.

    1. Finish the proof below by solving for a using a sequence of permitted algebraic manipulations.

      a, b R,
       
      a
      4
       
      =
       
      b + 3
      b
       


      implies

      # put proof steps here
      a = \undefined # should be an integer

    2. Finish the proof below by solving for b in terms of a.

      a,b R,
       
      11
      24
       
      =
       
      3
      7
       
      a +
       
      1
      2
       
      b

      implies
      # ???
      b = \undefined # should be an expression with "a"

    3. Finish the proof below to show that [1;2] and [-2;1] are linearly independent.

       
      1
      2
       
       
      -2
      1
       
      = 1 ⋅ -2 + 2 ⋅ 1   and

      # ???
      (
       
      1
      2
       
      ) and (
       
      -2
      1
       
      ) are orthogonal

    4. Solve for x (the solution is an integer).

      x R,
      (
       
      x
      4
       
      ) and (
       
      1
      1
       
      ) are orthogonal

      implies
      # ???
      x = \undefined

    5. Show that no [x; y] exists satisfying both of the below properties by deriving a contradiction (e.g., 1 = 0).

      x,y R,
      (
       
      x
      y
       
      ) is a unit vector   and
      (
       
      xx
      yy
       
      ) and (
       
      1
      1
       
      ) are orthogonal

      implies
      # ???
      \undefined # this should be a contradiction

  1. We have shown in lecture that R2, together with vector addition and scalar multiplication, satisfies some of the vector space axioms. In this problem, you will show that the remaining axioms are satisfied.
    1. Finish the proof below showing that [0; 0] is a left identity for addition of vectors in R2.

      x,y R,
      # ???
       
      0
      0
       
      +
       
      x
      y
       
      =
       
      x
      y
       


    2. Finish the proof below showing that [0; 0] is a right identity for addition of vectors in R2.

      x,y R,
      # ???
       
      x
      y
       
      +
       
      0
      0
       
      =
       
      x
      y
       


    3. Finish the proof below showing that 1 is the identity for scalar multiplication of vectors in R2.

      x,y R,
      # ???
      1 ⋅
       
      x
      y
       
      =
       
      x
      y
       


    4. Finish the proof below showing that the component-wise definition of inversion is consistent with the inversion axiom for vector spaces.

      x,y R,
      # ???
       
      x
      y
       
      + (-
       
      x
      y
       
      ) =
       
      0
      0
       


    5. Finish the proof below showing that the addition of vectors in R2 is associative.

      a,b,c,d,e,f R,
      # ???
       
      a
      b
       
      + (
       
      c
      d
       
      +
       
      e
      f
       
      ) = (
       
      a
      b
       
      +
       
      c
      d
       
      ) +
       
      e
      f
       


    6. Finish the proof below showing that the distributive property applies to vector addition and scalar multiplication for R2.

      s, x, y, x', y' R,
      # ???
      s (
       
      x
      y
       
      +
       
      x'
      y'
       
      ) = (s
       
      x
      y
       
      ) + (s
       
      x'
      y'
       
      )

  2. Any point p on the line between vectors u and v can be expressed as a(u-v)+u for some scalar a. In fact, it can also be expressed as b(u-v)+v for some other scalar b. In this problem, you will prove this fact for R3.

    Given some p = a(u-v)+u, find a formula for b in terms of a so that p = b(u-v)+v. Add this formula to the beginning of the proof (after a = ) and then complete the proof.

    a,b R, ∀ u,v,p R3,
    p
    =
    a(u-v) + u   and
    a
    =
    \undefined # define a in terms of b

    implies
    # ???
    p = b (u-v) + v

    The verifier's library contains a variety of derived algebraic properties for R and R3 in addition to the vector space axioms for R3. Look over them to see which might be useful.