Question

Find \(a_{0}, b_{0}, b_{1}, c_{0}, c_{1}\), and \(c_{2}\) such that the polynomials \(a_{0}, b_{0}+b_{1} t\), and \(c_{0}+c_{1} t+c_{2} t^{2}\) are mutually orthonormal in the interval \([0,1]\). The inner product is as defined for polynomials in Example \(2.2 .3\) with \(w(t)=1\).

Short answer

Therefore, the coefficients are \(a_{0}=1\), \(b_{0}=0\), \(b_{1}=\sqrt{3}\), \(c_{0}=0\), \(c_{1}=0\), and \(c_{2}=\sqrt{5}\).

Step-by-step solution

Polynomial \(a_{0}\)

Firstly, since this polynomial only has a constant term, the inner product of this polynomial with itself over the interval \([0,1]\) must be 1. Therefore, for \(b_{0}+b_{1} t\), the inner product would be 0. And for \(c_{0}+c_{1} t+c_{2} t^{2}\), the inner product would also be 0. The only value that satisfies these conditions is \(a_{0}=1\).

Polynomial \(b_{0}+b_{1} t\)

The inner product of this polynomial with itself over the interval \([0,1]\) should be 1. Also, the inner product of this polynomial with \(a_{0}=1\) should be 0, and the same goes for the polynomial \(c_{0}+c_{1} t+c_{2} t^{2}\). Solving these conditions gives \(b_{0}=0\), \(b_{1}=\sqrt{3}\).

Polynomial \(c_{0}+c_{1} t+c_{2} t^{2}\)

Through the same process, one can find that \(c_{0}=0\), \(c_{1}=0\) and \(c_{2}=\sqrt{5}\).

Key concepts

Inner Product of Polynomials

The concept of 'inner product' is a fundamental idea in linear algebra, and it is essential in understanding the orthogonality of polynomials. An inner product on a space of functions, such as polynomials, provides a way to measure angles and lengths within that space. Specifically for polynomials, we define the inner product over a certain interval, which in this case is \[0,1\], and with a weight function, here given as \(w(t)=1\).For two polynomials \(P(t)\) and \(Q(t)\), their inner product is given by the integral over the specified interval:\[ = \int_{0}^{1} P(t)Q(t)w(t)dt\]When the weight function is simply \(w(t) = 1\), this integral simplifies to the area under the curve of \(P(t)Q(t)\) from \(t = 0\) to \(t = 1\). For a polynomial to be normalized, its inner product with itself must equal one, which represents a 'unit length' in the polynomial space. This condition is crucial in finding the coefficients of orthonormal polynomials, like \(a_0\), as seen in the exercise.

Polynomial Orthogonality

Polynomial orthogonality goes hand in hand with the inner product, where two polynomials are orthogonal if their inner product is zero. This can be thought of as a generalization of perpendicular vectors in two- or three-dimensional space to the context of function spaces.In our context, for polynomials \(P(t)\) and \(Q(t)\) to be orthogonal on the interval \[0,1\], their inner product must satisfy:\[ = \int_{0}^{1} P(t)Q(t)w(t)dt = 0\]This signifies that there is no 'overlap' in the directions that \(P(t)\) and \(Q(t)\) point within the function space. In the provided exercise, the coefficients of the polynomials must be determined such that the polynomials are not only orthogonal to each other but also normalized. This requires solving for the coefficients by setting up a system of equations derived from the inner product conditions for orthogonality and normality, as demonstrated in Step 2 with polynomials \(b_0 + b_1 t\) and in Step 3 with \(c_0 + c_1 t + c_2 t^2\).

Mutually Orthonormal

When polynomials are 'mutually orthonormal,' it means every pair of different polynomials in the set is orthogonal, and each polynomial is normalized. For a set of polynomials, this entails that the inner product of any two distinct polynomials is zero, and the inner product of each polynomial with itself is one. Mutually orthonormal polynomials form a basis for the space of polynomials, where any other polynomial in that space can be expressed as a linear combination of this basis.For the exercise at hand, the goal is to find coefficients \(a_0, b_0, b_1, c_0, c_1\), and \(c_2\) such that the set \(\{a_0, b_0 + b_1 t, c_0 + c_1 t + c_2 t^2\}\) consists of mutually orthonormal polynomials. This condition places stringent requirements on the coefficients of the polynomials, leading to the unique solution provided in the steps. Such a set of polynomials has numerous applications, including solving differential equations, optimizing numerical integrations, and more, by exploiting their orthonormal properties.