Part d please (Coordinate systems, basis) Let \( V \) be the vector space \( \mathbb{P}_2 \) of polynomials of degree at most 2.(a) Use coordinate vectors to verify that the set \( \mathcal{B} = \{ 1, (t-1), (t-1)^2 \} \) forms a basis of \( \mathbb{P}_2 \).(b) Use coordinate vectors to verify that the set \( \mathcal{C} = \{ 1, (t+1), (t+1)^2 \} \) forms a basis of \( \mathbb{P}_2 \).(c) What are the coordinate vectors of \( 1, t+1 \) and \( (t+1)^2 \) relative to \( \mathcal{B} \)?(d) Find a matrix \( P \) such that for any polynomial \( f \in \mathbb{P}_2 \), we have \( P [f]_{\mathcal{B}} = [f]_{\mathcal{C}} \).

Let V be the vector space P2 of polynomials of degree at most 2. (a) The standard basis of P2 is given by: E={1,t,t2}. The given set is: B={1,t-1,t-12}. Find the coordinate vectors of elements of B relative to E. 1=1(1)+0(t)+0(t)2t-1=-1(1)+1(t)+0t2t-12=1(1)-2(t)+1t2. So, the coordinate vector are: p1E=100p2E=-110p3E=1-21 So, the matrix is: P=1-1101-2001. R1→R1+R2. P=10-101-2001 R2→R2+2R3R1→R1+R3. P=100010001 So, the matrix is in row echelon form. So, the vectors in B form a basis for P2. (b) The given set is: C={1,t+1,t+12}. Find the coordinate vectors of elements of C relative to E. 1=1(1)+0(t)+0(t2)t+1=1(1)+1(t)+0t2t+12=1(1)+2(t)+1t2. So, the coordinate vector are: p1E=100p2E=110p3E=121 So, the matrix is: P=111012001. R1→R1-R2. P=10-1012001 R2→R2-2R3R1→R1+R3. P=100010001 So, the matrix is in row echelon form. So, the vectors in C form a basis for P2. (c) Given vectors are: 1,t+1,t+12. Express them in terms of the vectors of B. 1=1(1)+0t-1+0t-12t+1=2(1)+1t-1+0t-12t+12=4(1)+4t-1+1t-12 So, the coordinate vectors are: p1B=100p2B=210p3B=441(d) Suppose, any polynomial in P2 be ft. So, ft=a0+a1t+a2t2. From part (c), the required matrix P is: P=124014001