Write your solution on separate sheets of paper and attach t

Write your solution on separate sheets of paper and attach them to this page before handing in. Show your work clearly. Is x(x - 1), (2x - 1)^2, 1 a spanning set for P_2? Are the matrices (1 1 1 1), (0 -1 1 0), (2 0 0 2), (0 1 -1 -1) linearly independent? It can be shown that the solutions to the differential equation y\" = y are exactly all functions of the form y = ae^x + be^-x, for constants a and b. (And it is not all that hard, either, but you do not need to do it.) These solutions form a vector space S = Span(e^x,e^-x). Show that cos h(x) = e^x + e^-x/2 and sin h (x) = e^x - e^-x/2 form a basis for S.

Solution

a)

x(x-1)               (2x-1)²                     1

the general polynomial in P₂ is

p(x)=a₀ p₁(x)+a₁p₂(x)+a₃p₃(x)

where p₁(x)=x(x-1)

p₂(x)=(2x-1)²

p₃(x)=1

p(x)=c₁(x)v(x-1)+c₂(2x-1)2++c₃

p(x)=x²(c₁+4c₂)-x(c₁+4c₂)+(c₂+c₃)

take p(x)=1+x+x²

^{comparing we get}

1=(c₁+4c₂) also c₁+4c₂=-1

which is impossible

so x(x-1),(2x-1)² and 1 are not spanning set  for p₂