Question

In \(\mathbf{P}_{3}\) find \(P_{D \leftarrow B}\) if \(B=\left\\{1, x, x^{2}, x^{3}\right\\}\) and \(D=\left\\{1,(1-x),(1-x)^{2},(1-x)^{3}\right\\}\). Then express \(p=a+b x+c x^{2}+d x^{3}\) as a polynomial in powers of \((1-x)\)

Short answer

\(P_{D \leftarrow B} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & -1 & -2 & -3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & -1 \end{bmatrix}\) and use its inverse to transform.

Step-by-step solution

Identify Polynomial Basis Change

To find the transition matrix \(P_{D \leftarrow B}\), we need to express each element of the basis \(D\) (\(\{1,(1-x),(1-x)^{2},(1-x)^{3}\}\)) in terms of the basis \(B\) (\(\{1, x, x^2, x^3\}\)).

Express Each Polynomial in Basis D

Let's express each element of \(D\) using \(B\):1. \(1 = 1\)2. \((1-x) = 1 - x\)3. \((1-x)^2 = 1 - 2x + x^2\)4. \((1-x)^3 = 1 - 3x + 3x^2 - x^3\)

Write Each Polynomial in Basis B

For each polynomial from Step 2, write them as column vectors in the basis \(B\):1. \(\begin{bmatrix} 1 \ 0 \ 0 \ 0 \end{bmatrix}\) for \(1\)2. \(\begin{bmatrix} 1 \ -1 \ 0 \ 0 \end{bmatrix}\) for \((1-x)\)3. \(\begin{bmatrix} 1 \ -2 \ 1 \ 0 \end{bmatrix}\) for \((1-x)^2\)4. \(\begin{bmatrix} 1 \ -3 \ 3 \ -1 \end{bmatrix}\) for \((1-x)^3\)

Construct Matrix \(P_{D \leftarrow B}\)

Combine the vectors from Step 3 to form the transition matrix \(P_{D \leftarrow B}\):\[P_{D \leftarrow B} = \begin{bmatrix}1 & 1 & 1 & 1 \0 & -1 & -2 & -3 \0 & 0 & 1 & 3 \0 & 0 & 0 & -1 \end{bmatrix}\]

Express Polynomial p in Basis D

Given \(p = a + bx + cx^2 + dx^3\), express it in the basis \(D\):Find the vector \(\begin{bmatrix} a \ b \ c \ d \end{bmatrix}\) in basis \(B\) using \(P_{D \leftarrow B}\) to transform it into basis \(D\):\[P_{D \leftarrow B}^{-1} \begin{bmatrix} a \ b \ c \ d \end{bmatrix} = \begin{bmatrix} a' \ b' \ c' \ d' \end{bmatrix}\]

Compute Inverse and Transformation

Calculate the inverse of \(P_{D \leftarrow B}\) and compute the transformed vector.Skipping inverse calculations for brevity, the inverse matrix can be applied to find:\[\begin{bmatrix} a' \ b' \ c' \ d' \end{bmatrix} = \begin{bmatrix} a \ -b-a \ c+2b+a \ -d-3c-3b-a \end{bmatrix}\]This represents the polynomial transformation.

Key concepts

Polynomial Basis Change

Polynomial basis change helps in expressing polynomials in different sets of basis. Basis refers to a set of functions relative to which a polynomial is expressed. For example, in vector spaces, a common set of bases is \({1, x, x^2, x^3}\). Each polynomial can be expressed as a linear combination of these basis functions.
In the given exercise, we have two bases: the standard basis \({1, x, x^2, x^3}\) represented by \(B\) and a different basis \(D\) which consists of \({1, (1-x), (1-x)^2, (1-x)^3}\). Transitioning from one basis to another means expressing the polynomials of one basis using the other. This transition is necessary for more flexible mathematical manipulations and simplifications.
Through polynomial basis change, polynomials become adaptable to various forms, opening many doors to efficient computation and fascinating algebraic insights.

Transition Matrix

The transition matrix, often denoted as \(P_{D \leftarrow B}\), is fundamental when transforming a polynomial from one basis to another. It maps the old basis to the new one, essentially telling us how to express the elements of one basis in terms of another.
Consider each element of the new basis \(D\). It can be expressed in terms of the standard basis \(B\). In our case:

• \(1\) maps to itself.
• \((1-x) = 1 - x\).
• \((1-x)^2 = 1 - 2x + x^2\).
• \((1-x)^3 = 1 - 3x + 3x^2 - x^3\).

These polynomial expressions are then structured as column vectors in terms of basis \(B\). The entire matrix \(P_{D \leftarrow B}\) is assembled by combining these vectors. It is this matrix that does the heavy lifting to transform polynomials, reflecting the comprehensive change from D to B. The matrix is crucial in multivariable calculus and other advanced mathematical scenarios.

Polynomial Expression in Different Basis

A polynomial can take various forms depending on the basis it is expressed in. Converting a polynomial to a different basis provides unique insights and potentially simpler forms.
In our exercise, the polynomial \(p = a + bx + cx^2 + dx^3\) is expressed in the original basis \(B\). However, it becomes necessary to express the polynomial in the basis \(D\), which consists of powers of \( (1-x) \). This involves transforming \(p\) using the transition matrix so that it looks like \(p = a' + b'(1-x) + c'(1-x)^2 + d'(1-x)^3\).
By finding the inverse of the transition matrix \(P_{D \leftarrow B}\), you can transform the vector representation of \(p\) from a basis \(B\) form to a basis \(D\) form, thereby rewriting the polynomial in our desired powers. This method helps simplify the polynomial expression by considering different variables and coefficients, revealing distinct algebraic properties and applications.