Polynomial Proof

This proof was written a while ago, when I read a note with an interesting question:

Define polynomials in this way:


\(P_n(x) = \frac{x(x-1)(x-2)...(x-n)}{n!}\) where \(n \in \mathbb{Z}^{+}\)

Prove that for any polynomial \(f(x)\) \(\exists\) unique real numbers \(b_0,b_1,...,b_n\) such that:

\(f(x)=\displaystyle \sum_{0\leq i \leq n} b_ip_i(x)\)

The proof for this is purely based in Linear Algebra and assumes the reader is familiar with the concepts of a determinate, invertible matrix, linear system, etc.


Let \( f(x)= c_0 +c_1x +c_2x^2+...+c_nx^n\) and take note of the fact that:

\(p_n(x)= \dbinom{x}{n}(x-n)\). Since \(p_0(x)=1\), we may write:

\(\displaystyle \sum_{0 \le i \le n}(b_i)(p_i(x)) = (b_0)+\displaystyle \sum_{1 \le i \le n}(b_i)(p_i(x))\)

Expanding \(\displaystyle \sum_{1 \le i \le n}(b_i)(p_i(x))\):

(Without loss of generality, assume \(\displaystyle \sum_{1 \le i \le n}(b_i)(p_i(x))\)= \(\displaystyle \sum_{i=1}^{n}(b_i)(p_i(x))\))

\(\displaystyle \sum_{i=1}^{n}(b_i)(p_i(x))\))= \(b_1(x^2-x)+\frac{b_2}{2}(x^3-3x^2+2x)+\frac{b_3}{6}(x^4-6x^3+11x^2-6x)+\)

\(\frac{b_4}{24}(x^5-10x^4+35x^3-50x^2+24x)+... +\frac{b_nx!(x-n)}{n!(x-n)!}\)

Equating coefficients of f(x), \(c_i\), with those of \(\displaystyle \sum_{i=1}^{n}(b_i)(p_i(x))\)):


\(c_1=-b_1+b_2-b_3+b_4+....=\displaystyle \sum_{i=1}^{n}(-1)^ib_i\)


\(c_3= \frac{b_2}{2} - b_3 +\frac{35}{24}b_4-....\)

\(c_4= \frac{b_3}{6} - \frac{10}{24}b_4+.....\)

\(c_5= \frac{b_4}{24}+....\)

This is a linear system of n equations (excluding \(c_0=b_0\)) in n unknowns. Therefore, we may construct an n x n matrix A such that: \(A\vec{b}=\vec{c}\), where \(\vec{b}\) contains all \(b_i\) and \(\vec{c}\) contains all \(c_i\). \(A=\)

\[ \begin{pmatrix} -1 & 1 & -1&...&\alpha_1 \\ 1 & \frac{-3}{2} & \frac{11}{6}&...&\alpha_2 \\ 0 & \frac{1}{2} & -1&...&\alpha_3\\ 0&0&\frac{1}{6}&...&\alpha_4\\ .&.&.&.&.\\ .&.&.&.&.\\ .&.&.&.&.\\ 0&0&0&....&\alpha_n\\ \end{pmatrix} \]

\(A\vec{b}=\vec{c}\) has a unique solution if and only if A is an invertible matrix. A is invertible if and only if \(det(A)\neq 0\). Then \(A\vec{b}=\vec{c}\) has a unique solution if and only if \(det(A)\neq 0\). If we use Gaussian Elimination to reduce A, that is, if we add line (1) to line (2), then line (2) to line (3), then line (3) to line (4), etc. we see that A will be reduced to an upper triangular matrix with every term on the main diagonal of A of the form: \(A_{ji}=\frac{-1}{i!}\), where \(i\) corresponds to the column number of \(A_{ji}\). Hence \(det(A)=\displaystyle \prod_{i=1}^n\frac{-1}{i!}\).

Since \(det(A)=\displaystyle \prod_{i=1}^n\frac{-1}{i!}\) and \(i\ge1\), \(det(A)\neq 0\). Then A is invertible and \(A\vec{b}=\vec{c}\) has a unique solution, hence \(\vec{b}=A^{-1}\vec{c}\), which of course exists. This proves the existence of \(b_i\) but does not prove the uniqueness of all \(b_i\). We will now proceed to prove this.

Let \(A_{ji}^{-1}\) denote the element of \(A^{-1}\) in the \(j^{th}\) row and \(i^{th}\) column. Since \(\vec{b}=A^{-1}\vec{c}\), we have:

\(b_n=A_{n1}^{-1}c_1 + A_{n2}^{-1}c_2+A_{n3}^{-1}c_3+...+A_{nn}^{-1}c_n\)

Suppose \(\exists\) \(b_i\) such that \(b_i=b_n\) but \(i \neq n\), i.e. suppose all \(b_i\) are not unique. Then, since \(b_n\) is described by the above equation:

\( A_{i1}^{-1}c_1=A_{n1}^{-1}c_1\)

\( A_{i2}^{-1}c_2=A_{n2}^{-1}c_2\)

\( A_{i3}^{-1}c_3=A_{n3}^{-1}c_3\)

And in general: \( A_{in}^{-1}c_n=A_{nn}^{-1}c_n\) \(\Rightarrow\) \( A_{in}^{-1}=A_{nn}^{-1}\)

If \( A_{in}^{-1}=A_{nn}^{-1}\) and \(i \neq n\), then \(A^{-1}\) contains at least two identical rows, which implies \(det(A^{-1})=0\). Then \(A^{-1}\) is not invertible, which is impossible because \((A^{-1})^{-1}=A\). This is a contradiction, hence we are forced to conclude that all \(b_i\) are unique.


It is also interesting to note that as \(n\) approaches infinity, \(det(A)\) will approach zero. This implies that as the degree of \(f(x)\) becomes increasingly large, the constants \(b_i\) will become increasingly "precise," that is to say they will most likely become irrational, or rational with an increasingly large denominator. This makes sense, because when \(n\) becomes large, the values of \(b_i\) required to satisfy our linear system become very specific.

Note by Ethan Robinett
3 years, 8 months ago

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Why don't we just show that the polynomials \(P_i(x), 0\leq i \leq n\) are linearly independent, and invoke the vector space theory for polynomials of degree \(n\) ?

Abhishek Sinha - 3 years, 8 months ago

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Essentially, this proof rests upon the fact that the matrix A is composed of linearly independent vectors (which is of course the reason why \(det(A) \neq 0\)). If you examine what actually lead to the formation of A, you'll see that A is simply an array of the coefficients of \(x^n\) produced by \(\displaystyle \sum_{0 \leq i \leq n} b_ip_i(x)\). So what we proved is that . \(\displaystyle \sum_{0 \leq i \leq n} b_ip_i(x)\) produces a linear map A which transforms \(\vec{b}\) to \(\vec{c}\) in, of course, a different basis. I believe what I did here is a more detail-oriented presentation of what you just said.

Ethan Robinett - 3 years, 8 months ago

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Yes, but this is evident without doing any calculations. Just note that the polynomials \(P_i(x)\) have increasing degrees and hence they are linearly independent.

Abhishek Sinha - 3 years, 8 months ago

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@Abhishek Sinha I see what you're saying. I guess when I wrote this I figured that just saying something to that effect wouldn't be sufficient to prove the statement. I definitely see what you mean though

Ethan Robinett - 3 years, 8 months ago

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