Question

Let $\left\{\alpha_0, \ldots, \alpha_n\right\}$ and $\left\{\beta_0, \ldots, \beta_n\right\}$ be two sets of points in $\mathbb{C}$. Show that if $\alpha_i \neq \alpha_j$ when $i \neq j$, then there exists a unique polynomial $p(\lambda)=c_0+c_1 \lambda+\cdots+c_n \lambda^n$ of degree $n$ such that $p\left(\alpha_i\right)=\beta_i$ for $i=0, \ldots, n$ and find a formula for the coefficients $c_j$.

   Let $\left\{\alpha_0, \ldots, \alpha_n\right\}$ and $\left\{\beta_0, \ldots, \beta_n\right\}$ be two sets of points in $\mathbb{C}$. Show that if $\alpha_i \neq \alpha_j$ when $i \neq j$, then there exists a unique polynomial $p(\lambda)=c_0+c_1 \lambda+\cdots+c_n \lambda^n$ of degree $n$ such that $p\left(\alpha_i\right)=\beta_i$ for $i=0, \ldots, n$ and find a formula for the coefficients $c_j$.

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Linear Algebra in Action

Linear Algebra in Action

Harry Dym 1st Edition

Chapter 5, Problem 42

Instant Answer

Step 1

Since there are $n+1$ conditions (one for each $i$), and we are looking for a polynomial of degree $n$, this is a classic polynomial interpolation problem. Show more…

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Let $\left\{\alpha_0, \ldots, \alpha_n\right\}$ and $\left\{\beta_0, \ldots, \beta_n\right\}$ be two sets of points in $\mathbb{C}$. Show that if $\alpha_i \neq \alpha_j$ when $i \neq j$, then there exists a unique polynomial $p(\lambda)=c_0+c_1 \lambda+\cdots+c_n \lambda^n$ of degree $n$ such that $p\left(\alpha_i\right)=\beta_i$ for $i=0, \ldots, n$ and find a formula for the coefficients $c_j$.

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