Try the fastest way to create flashcards
Question

# (a) Prove that there exist unique scalars $c_0,c_1,\dots,c_n$ such that$f(-1)+f(-2)=c_0f(0)+c_1f(1)+\dots+c_nf(n)$for every polynomial $f(x)$ in $\mathcal{P}_n$. (b) Find the scalars $c_0,c_1,c_2,c_3,$ and $c_4$ such that$f(-1)+f(-2)=c_0f(0)+c_1f(1)+c_2f(2)+c_3f(3)+c_4f(4)$for every polynomial $f(x)$ in $\mathcal{P}_4$.

Solution

Verified
Step 1
1 of 4

#### (a)

According to $\textit{Exercise 77}$, there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$\int\limits_{-2}^{-1}f(x)dx=c_0f(0)+c_1f(1)+\dots+c_nf(n)$

for every polynomial $f(x)$ in $\mathcal{P}_n$. The expression on the left is a number determined uniquely by a polynomial $f(x)$, if we switch this expression with $f(-1)+f(-2)$, this is still determined uniquely by the polynomial $f(x)$ and it doesn't change the statement.

Therefore, there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$f(-1)+f(-2)=c_0f(0)+c_1f(1)+\dots+c_nf(n)$

for every polynomial $f(x)$ in $\mathcal{P}_n$.

## Recommended textbook solutions #### Elementary Linear Algebra: A Matrix Approach

2nd EditionISBN: 9780131871410 (2 more)Arnold Insel, Lawrence Spence, Stephen Friedberg
3,813 solutions #### Linear Algebra and Its Applications

5th EditionISBN: 9780321982384 (3 more)David C. Lay, Judi J. McDonald, Steven R. Lay
2,070 solutions #### Elementary Linear Algebra

11th EditionISBN: 9781118473504Howard Anton
2,932 solutions #### Elementary Linear Algebra

12th EditionISBN: 9781119406778Anton Kaul, Howard Anton
3,078 solutions