Coefficient Maps Between Polynomial Spaces and R^n

The Coefficient Map and Its Basis

Polynomials of bounded degree can be represented by finite lists of real numbers: their coefficients. To turn that observation into an isomorphism, the order of those coefficients must be fixed, and the map must be checked to be both linear and bijective. Here the relevant order comes from the monomials \(1,x,\dots,x^{n-1}\), rather than from the powers in descending order.

Let \(n\in\mathbb{N}\). Write \(\mathcal{P}_{n-1}(\mathbb{R})\) for the real vector space of polynomials of degree at most \(n-1\), including the zero polynomial. Every \(p\in\mathcal{P}_{n-1}(\mathbb{R})\) has a unique expression \(p(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}\), where some of the coefficients may be zero. Polynomials are treated as formal expressions, so equality of two polynomials means that their corresponding coefficients are equal.

The coefficient vector is a column, with the constant coefficient first. In terms of coordinates relative to \(\mathcal{B}\), this definition says \([\ p\ ]_{\mathcal{B}}=\Phi(p)\). The entries of \(\Phi(p)\) are not values of \(p\) at selected inputs; they are the scalars multiplying the ordered monomials.

Why the Map Is an Isomorphism

A linear map preserves addition and scalar multiplication. A map is an isomorphism if it is linear and bijective, meaning it is both one-to-one and onto. We verify these requirements directly, rather than relying only on the visual similarity between a polynomial and its list of coefficients.

Proof. First let \(p(x)=\sum_{k=0}^{n-1}a_kx^k\) and \(q(x)=\sum_{k=0}^{n-1}b_kx^k\) be in \(\mathcal{P}_{n-1}(\mathbb{R})\), and let \(\alpha,\beta\in\mathbb{R}\). The coefficient of \(x^k\) in \(\alpha p+\beta q\) is \(\alpha a_k+\beta b_k\), for each \(k=0,\dots,n-1\). Hence \(\Phi(\alpha p+\beta q)=(\alpha a_0+\beta b_0,\dots,\alpha a_{n-1}+\beta b_{n-1})^\top=\alpha\Phi(p)+\beta\Phi(q)\). Thus \(\Phi\) is linear.

Now define \(\Psi\) by the formula in the theorem. Its output has degree at most \(n-1\), so \(\Psi\) maps into \(\mathcal{P}_{n-1}(\mathbb{R})\). For any \(p(x)=\sum_{k=0}^{n-1}a_kx^k\), the definition of \(\Phi\) gives \(\Psi(\Phi(p))=\sum_{k=0}^{n-1}a_kx^k=p\). For any \(\mathbf{c}=(c_0,\dots,c_{n-1})^\top\in\mathbb{R}^n\), the definition of \(\Psi\) followed by coefficient extraction gives \(\Phi(\Psi(\mathbf{c}))=(c_0,\dots,c_{n-1})^\top=\mathbf{c}\). Therefore \(\Psi\) is a two-sided inverse of \(\Phi\), so \(\Phi\) is bijective. Together with linearity, this proves that \(\Phi\) is an isomorphism. \(\square\)

The two inverse identities do separate jobs. The equation \(\Psi(\Phi(p))=p\) confirms that no two polynomials are sent to the same column; \(\Phi(\Psi(\mathbf{c}))=\mathbf{c}\) confirms that every column in \(\mathbb{R}^n\) occurs. Establishing both avoids confusing one-to-one with onto.

What the Matrix of the Map Looks Like

The matrix of a linear map depends on the chosen ordered bases. For a map from \(\mathcal{P}_{n-1}(\mathbb{R})\) to \(\mathbb{R}^n\), use the monomial basis \(\mathcal{B}=(1,x,\dots,x^{n-1})\) in the domain and the standard basis in the target. The columns of the representing matrix are the images of the domain basis vectors, written in target coordinates.

Proof. For each \(j=1,\dots,n\), the \(j\)th vector of \(\mathcal{B}\) is \(x^{j-1}\). Its coefficient is \(1\) in position \(j\) and \(0\) in every other position. Consequently \(\Phi(x^{j-1})=\mathbf{e}_j\). The \(j\)th column of the representing matrix is therefore \(\mathbf{e}_j\). Since this holds for all \(j=1,\dots,n\), the matrix has the standard basis vectors as its columns, which is exactly \(\mathbf{I}_n\). \(\square\)

The identity matrix here depends on the bases just specified. It does not mean that the input polynomial is literally a vector in \(\mathbb{R}^n\); the polynomial and the column are elements of different vector spaces. The map sends each ordered basis vector in the domain to the corresponding standard basis vector in the target. Thus the identity matrix records the coordinate correspondence, while the map itself remains a function between polynomial and coordinate spaces.

Common Pitfall: Reversing the Coefficient Order

For the monomial basis \((1,x,\dots,x^{n-1})\), coefficients must be listed from the constant term upward. A reversed list can describe a different, perfectly valid coordinate convention, but it is not the coefficient map defined here unless the basis order is reversed too.

For example, take \(p(x)=4-2x+7x^2\) in \(\mathcal{P}_2(\mathbb{R})\). The correct coefficient column for the ordered basis \((1,x,x^2)\) is \(\Phi(p)=(4,-2,7)^\top\). A common incorrect calculation is to read coefficients from the highest power down and write \((7,-2,4)^\top\) as the image under \(\Phi\). Applying the inverse map to that incorrect column produces \(\Psi((7,-2,4)^\top)=7-2x+4x^2\), which is not \(4-2x+7x^2\). The correction is to match each entry with its basis monomial: \(4\) multiplies \(1\), \(-2\) multiplies \(x\), and \(7\) multiplies \(x^2\). Then \(\Psi((4,-2,7)^\top)=4-2x+7x^2=p(x)\), as required.

This example also explains why basis order is part of the definition rather than a cosmetic choice. If the domain basis were instead \((x^2,x,1)\), then the reversed column would be appropriate coordinates for that different basis. A coordinate column is meaningful only when its basis and order are known.

Using the Correspondence

The coefficient map gives a direct translation between polynomial questions and vector questions, provided the translation uses the same order throughout. Constructing the map requires identifying the allowed powers, filling in any missing coefficients with zeros, and placing the coefficients in a column. Returning to a polynomial requires matching each entry back to its monomial. The inverse proof guarantees that neither translation loses information.

For example, the zero polynomial maps to \(\mathbf{0}\), since all its coefficients are zero. A constant polynomial \(c\) in \(\mathcal{P}_{n-1}(\mathbb{R})\) maps to \((c,0,\dots,0)^\top\); the final entries remain present even though the higher powers do not occur. The identity matrix representation is useful because calculations with coordinates can be carried out in \(\mathbb{R}^n\), then translated back with \(\Psi\). The interpretation of the resulting column still depends on its polynomial basis.