$S = \{t^3+t^2, t^2+t, t+1, 1\}$ is a basis of $P_3(t)$. Find the coordinate vector $[v]$ of $v$ relative to $S$ where:
(a) $\quad v = 2t^3+t^2-4t+2$
(b) $\quad v = at^3+bt^2+ct+d$
I'm finding, for (a), that the coordinate vector $$ [v] = \begin{bmatrix} 2 \\ -1 \\ -3 \\ 5 \end{bmatrix}, $$ but the answer in Seymour Lipschutz's Linear Algebra book is $$ [v] = \begin{bmatrix} 2 \\ -1 \\ 2 \\ 2 \end{bmatrix}. $$
For this question, I found that $$ at^3 + (a+b)t^2 + (b+c)t + (c+d) = 2t^3 + t^2 - 4t + 2. $$ So, $a = 2$, then $b = -1$, $c = -3$, and $d = 5$. What did I do wrong??
And for part (b), I'm finding that $$ [v] = \begin{bmatrix} a \\ b - a \\ c - b + a \\ d - c + b - a \end{bmatrix}, $$ but the answer in the same book is $$ [v] = \begin{bmatrix} a \\ b - c \\ c - b + a \\ d - c + b - a \end{bmatrix}, $$ with the difference being only the $b-a$ to $b-c$. Is the book wrong? Please help me, I'm stuck :(