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Quiz

Proof by magic 1

JAV Hns 9 questions

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Questions in this quiz (9)

  1. 1Are direct sums of vector spaces unique?
    1. AYes
    2. BUp to isomorphism
    3. CNo
  2. 2If V, X are direct sums of U,W;
    with a:U->V, b:W->V, c:U->X, d:W->X
    If a,b,c,d are linear then...
    1. AA map between V,X would be unique
    2. BA map between V,X would not nec. be unique
    3. CA map R:V->X would have aoR=c
    4. DA map R:V->X would have Roa=c
    5. EA map R:V->X would have boR=d
    6. FA map R:V->X would have Rob=d
  3. 3Let V be an IPS then for all ƒ in V*
    1. Athere is a UNIQUE v in V s.t. ƒ={v,.}...
    2. B...provided that V is finite
    3. C...provided that V is finite dimensional
    4. Dthere exists a v in V s.t. ƒ={v,.}... [not nec. unique]
    5. Ethere is a UNIQUE v in V s.t. ƒ={.,v}...
    6. Fthere exists a v in V s.t. ƒ={.,v}... [not nec. unique]
    7. Gp:v |->{v,.} is a linear bijection if V a v.s. over ®
    8. Hp:v |->{v,.} is a conjugate linear bijection if V a v.s. over ©
    9. Iƒ is not a bijection
    10. Jƒ is not linear
  4. 4Given that {v,w}=0 for all w in a vector space V, there may be more than one possible v that fits
    1. ATrue
    2. BFalse
  5. 5If the perpendicular space a set of v in V s.t.
    in each case their inner product with
    every element of a subspace U of V
    is equal to zero
    1. ATrue
    2. BFalse
  6. 6If U,W are orthogonal subspaces of V, then...
    1. AUnW = {0}
    2. BUnW = ø
    3. CU+W=U(+)W
    4. DUxW=U(x)W
    5. EdimU + dimW = dimV
      if V finite dimensional
    6. F(dimU + dimW) = dimV|u+w
  7. 7A LINEAR operator can have
    two distinct bases
    for it's domain and codomain
    1. ATrue
    2. BFalse
  8. 8ƒ,¥ are adjoint linear operators means that...
    1. A{ƒ(v),w} = {v,¥(w)}
    2. Bƒ(v)o¥(v)=0
    3. Cƒo¥ = id
  9. 9ƒ a linear operator
    acting on a finite dimensional IPS, V
    If B is an orthonormal basis for V
    And A is the matrix for ƒ w.r.t. B, then...
    1. Aƒ has an adjoint, ¥
    2. Bƒ doesn't necessarily have an adjoint ¥
    3. Cthe conjugate transpose of A, A^† is the matrix representing the inverse map to ƒ
      w.r.t. B
    4. Dthe conjugate transpose of A, A^† is the matrix representing the adjoint map to ƒ
      w.r.t. B

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Description
A WitchcraftA Quiz on Proof by magic 1, created by JAV Hns on 28/12/2015.
Author
JAV Hns
Language
English
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