MATA22 Assignment 3

MATA22 Assignment 3
Due Date: On the week March 2-6 in your own tutorials at rst 15 mins

  1. This question helps you understand subspace and basis
    (a) Let U = f(z1; z2; z3; z4; z5) 2 C : 6z1 = z2; z3 + 2z4 + 3z5 = 0g.
    Check if U is a subspace. If U is a subspace, then nd a basis
    which span U.
    (b) Suppose b1; b2; b3; b4 is a basis of vector space V . Prove that
    b1 + b2; b2 + b3; b3 + b4; b4
    is also a basis of V:
    (c) Prove or disprove : there exist a basis p0; p1; p2; p3 2 P3(R) ( P3(R)
    is the set of all polynomial over real number with highest possible
    degree = 3 ) such that none of polynomials p0; p1; p2; p3 has degree

    (d) Suppose U and W are both ve-dimensional subspaces of R9. Prove
    that U \W 6= f0g:
  2. This question helps you understand direct sum
    (a) Suppose U1; U2; U3:::Un are nite-dimensional subspaces of V such
    U1 + U2 + U3 + ::: + Un
    is a direct sum. Prove that
    U1  U2  U3  :::  Un
    is also nite-dimensional and
    dim(U1  U2  U3  :::  Un) = i=n
    (b) Let
    Ue = ff(x) : R ??! Rjf(??x) = f(x)g
    Uo = ff(x) : R ??! Rjf(??x) = ??f(x)g
    i. Prove Ue; Uo are subspaces.
    ii. Prove any functions f(x) : R ??! R ( denote as RR ) can be
    represented as sum of an even function and an odd function.
    iii. Prove that
    RR = Ue  Uo
    by using above facts.
    (c) Check if the following statement is true. If it’s true, prove it. If not
    , show a counter example.
    fMnng = fSynng  fSknng
    where fSynng means the set of all n by n symmetric matrices and
    fSknng means the set of all n by n skew symmetric matrices.
  3. This question helps you understand linear map
    (a) Suppose a; b 2 R: De ne T : R3 ??! R2 by
    T(x; y; z) = (2x ?? 4y + 3z + b; 6x + cxyz):
    i. show that T is linear map if and only if a = b = 0.
    ii. Can you nd the matrix representation of T, if a = b = 0.
    iii. Find the range and kernel of T, if a = b = 0.
    (b) Can you give an example of an isomorphism mapping from R3 to
    P2(R) ?