Question

Let \(a\) and \(b\) denote distinct numbers. a. Show that \(\\{(x-a),(x-b)\\}\) is a basis of \(\mathbf{P}_{1}\). b. Show that \(\left\\{(x-a)^{2},(x-a)(x-b),(x-b)^{2}\right\\}\) is a basis of \(\mathbf{P}_{2}\). c. Show that \(\left\\{(x-a)^{n},(x-a)^{n-1}(x-b)\right.\), \(\left.\ldots,(x-a)(x-b)^{n-1},(x-b)^{n}\right\\}\) is a basis of \(\mathbf{P}_{n}\)

Short answer

Each set of polynomials form a basis for their respective polynomial space, demonstrated by linear independence and spanning arguments.

Step-by-step solution

Understanding Polynomial Spaces

Before solving, let's understand that \( \mathbf{P}_n \) denotes the space of polynomials of degree \( n \). A basis here is a set of linearly independent polynomials that span the entire space \( \mathbf{P}_n \).

Basis of \( \mathbf{P}_1 \)

The space \( \mathbf{P}_1 \) consists of polynomials of degree at most 1, which can be written as \( p(x) = c_0 + c_1x \). The polynomials \( (x-a) \) and \( (x-b) \) are linear functions — both are degree 1. They are linearly independent because if \( c_1(x-a) + c_2(x-b) = 0 \), then setting \( x = a \) and \( x = b \) will individually imply \( c_2 = 0 \) and \( c_1 = 0 \), making \( c_1 = c_2 = 0 \). Hence, they form a basis of \( \mathbf{P}_1 \).

Basis of \( \mathbf{P}_2 \)

\( \mathbf{P}_2 \) includes polynomials up to degree 2. Consider \( (x-a)^2, (x-a)(x-b), (x-b)^2 \). These are degree 2 polynomials, and they are linearly independent because none can be written as a combination of the others (substituting suitable values for \( x \) shows zero coefficients). They also span \( \mathbf{P}_2 \), as any quadratic polynomial like \( ax^2 + bx + c \) can be written in terms of these three polynomials using combinations of specific values.

Basis of \( \mathbf{P}_n \)

The expression \( \{(x-a)^n, (x-a)^{n-1}(x-b), \ldots, (x-b)^n \} \) can be shown to be a basis for \( \mathbf{P}_n \) using an induction argument. The polynomials are of increasing degree from 0 to \( n \) based on the binomial expansion of polynomials. They are linearly independent for the same reason as simpler degrees — substituting specific values for \( x \) yields a system like the Vandermonde matrix, which is non-singular given distinct roots \( a \) and \( b \). So, these polynomials maintain independence and can express any polynomial of degree \( n \), thus forming a basis.

Key concepts

Linear Independence

In mathematics, the notion of linear independence is crucial when discussing sets of polynomials or vectors. A set of polynomials is considered linearly independent if no polynomial in the set can be expressed as a linear combination of the others. This means that if you were to try to write one polynomial in terms of the others, you would not be able to do so without introducing non-zero coefficients that multiply each polynomial - unless all those coefficients are zero.

For example, consider the polynomials \((x-a)\) and \((x-b)\). We say they are linearly independent because the only solution to the equation \(c_1(x-a) + c_2(x-b) = 0\) is \(c_1 = 0\) and \(c_2 = 0\), given that \(a\) and \(b\) are distinct. Linear independence allows for the creation of a basis within polynomial spaces, ensuring that the space is efficiently spanned without redundancy.

• Each polynomial contributes uniquely to the structure of the space.
• Linear independence hints that no polynomial in the basis can be recreated by weighing another in the basis set.

Polynomial Spaces

Polynomial spaces, denoted by \(\mathbf{P}_n\), are mathematical constructs representing all polynomials of degree \(n\) or less. For example, \(\mathbf{P}_1\) consists of all linear polynomials, such as \(p(x) = c_0 + c_1x\), where \(c_0\) and \(c_1\) are coefficients. The space \(\mathbf{P}_2\) includes quadratic polynomials like \(ax^2 + bx + c\).

These spaces are important because they help mathematicians understand the behavior of polynomials by grouping them into easily manageable sets. Each space has a dimension corresponding to \(n + 1\), meaning that \(\mathbf{P}_1\) has dimension 2, and \(\mathbf{P}_2\) has dimension 3, and so on. The dimension indicates the number of polynomials needed to form a basis for that space.

• Polynomial spaces organize polynomials based on their degree.
• The dimension guides how many basis polynomials are necessary.

Degree of Polynomials

The degree of a polynomial is defined as the highest power of the variable present in the polynomial expression. For instance, in the polynomial \(3x^2 + 2x + 1\), the degree is 2 because the term \(x^2\) has the highest power.

Understanding the degree is vital because it dictates the polynomial's behavior, including how many roots it can have and its curvature on a graph. Moreover, when identifying a set of polynomials as a basis for a polynomial space, having distinct degrees aids in ensuring they can span the full space effectively.

When constructing bases like \((x-a)^n\) or \((x-b)^n\), varying the degrees ensures that every polynomial in the basis contributes uniquely to spanning the space. For instance, in \(\mathbf{P}_n\), you seek polynomials of degree up to \(n\), leveraging those degrees to form a comprehensive basis.

• The degree informs the potential roots and visual graph behavior of the polynomial.
• Distinct degrees across a polynomial set help in creating an efficient basis.