## 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
that
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
i=1dim(Ui):
(b) Let
Ue = ff(x) : R ??! Rjf(??x) = f(x)g
and
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.
fMnng = fSynng  fSknng
where fSynng means the set of all n by n symmetric matrices and
fSknng 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) ?