Help for this class engineering mathematics show all step pl

Help for this class engineering mathematics

show all step please!!

Let P_2 be the vector space consisting of all polynomials with real coefficients of degree less than or equal to 2. Determine whether each of the following sets is a basis for P_2. Justify your answer in each case. A = {1 + t, t + t^2}. B = {1, -1 + t, 1 -2t + t^2}. Choose a set from part (a) that is a basis for P_2. Find the coordinates of the polynomial p(t) = 6 - 5t + 2t^2 relative to that basis.

Solution

a)

i)

P2 has dimension 3 so A cannot be a basis for P2 because A has only two vectors in it

ii)

1+(-1+t)=t

HEnce, 1 and t belongs to span (B)

1-2t+t^2-1+2*t=t^2

HEnce, 1,t,t^2 belongs to span (B)

Hence, the three vectors span P2

Hence, B is a basis for P2

b)

B forms basis for we will find coordinates of p(t) w.r.t. B

LEt, a,b,c be the coordinates

Then, a+b(-1+t)+c(1-2t+t^2)=6-5t+2t^2

a-b+c+(b-2c)t+ct^2=6-5t+2t^2

comparing coefficeints of t,t^2 and constant terms gives

a-b+c=6

c=2

b-2c=-5

b=-5+2c=-1

b=-1

a-b+c=6

a+1-2=6

a=7

Hence coordinates of p(t) are

(7,-1,2)