MAT224 ASSIGNMENT 1
DUE BY FRIDAY MAY 22, 2020, 11:59 PM
Each question is worth 5 marks.
Question 1. Let V = R2. For (a1,a2),(b1,b2) ∈ V and c ∈ R define
(0, 0) if c = 0 (a1,a2)+(b1,b2)=(a1+b1,a2+b2)andc(a1,a2)= ca1,a2 ifc̸=0.
c
Is V a vector space with these operations? If yes, prove it. If not, give an example showing one of the axioms fails.
Question2.LetV beavectorspace.Showthatx=vistheonlysolutiontotheequationx+x=2vinV.Citeall axioms used.
Question 3. Suppose n ≥ 1, and let Un = {xg(x) : g(x) ∈ Pn−1}. Is Un a subspace of Pn? If yes, prove it. If not, find a counter example.
Question 4. (a) Suppose U and W are subspaces of a vector space V . Prove that U ∩ W = {v : v ∈ U and v ∈ W } is a subspace of V .
(b)GiveanexampleoftwosubspacesU andW andavectorspaceV suchthatU∪W ={v:v∈U orv∈W}isnot a subspace of V .
1
Question 5. Let U = f ∈ F[0,1] : f is integrable and
answer. Note: You can use without justification your knowledge from first year calculus.
1
0
f(x)dx = 0 . Is U a subspace of F[0,1]? Justify your