Question

If \(\mathbf{P}_{n}=\operatorname{span}\left\\{p_{1}(x), p_{2}(x), \ldots, p_{k}(x)\right\\}\) and \(a\) is in \(\mathbb{R}\), show that \(p_{i}(a) \neq 0\) for some \(i\).

Short answer

For some \(i\), \(p_i(a) \neq 0\); otherwise, it contradicts the polynomial space properties.

Step-by-step solution

Understanding the Problem

We are given a vector space \(\mathbf{P}_{n}=\operatorname{span}\{p_{1}(x), p_{2}(x), \ldots, p_{k}(x)\}\), which means that any polynomial \(f(x)\) in \(\mathbf{P}_{n}\) can be expressed as a linear combination of \(p_1(x), p_2(x), \ldots, p_k(x)\). We need to show that for at least one polynomial in the basis, the evaluation at \(a\) is non-zero — \(p_i(a) eq 0\).

Suppose the Contrary

Assume that for all \(i\), \(p_i(a)=0\). This implies that every polynomial \(p_i(x)\) in the span equals zero when evaluated at \(x = a\).

Implication of Non-Zero Polynomial

If \(p_i(a) = 0\) for all \(i\), then any polynomial formed by a linear combination of these \(p_i(x)\) will also be zero when evaluated at \(a\). Thus, every polynomial \(f(x)\) in \(\mathbf{P}_{n}\) would satisfy \(f(a) = 0\).

Contradiction with Polynomial Space

A key property of any polynomial space spanned by non-zero polynomials is that there exists a polynomial of degree zero, usually 1, which does not vanish anywhere on \(\mathbb{R}\). If \(f(a)=0\) for all \(f(x)\) in \(\mathbf{P}_{n}\), even the constant polynomial function 1 must also evaluate to 0, which is a contradiction since 1 is never zero.

Conclusion

Since assuming \(p_i(a) = 0\) for all \(i\) leads to a contradiction, our initial assumption must be false. Thus, there exists at least one \(p_i(x)\) for which \(p_i(a) eq 0\).

Key concepts

Polynomials

A polynomial is an expression comprising variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Polynomials can be a single constant term, a linear equation, or even a more complex expression such as a cubic polynomial. They serve as fundamental expressions in algebra that help interpret natural phenomena through equations.

Features of polynomials include:

• They are made up of terms, each being a product of a coefficient and a power of a variable.
• The degree of a polynomial is determined by the highest power of the variable in the expression.
• Polynomials can be classified based on their degrees: linear (degree 1), quadratic (degree 2), and so forth.

In the context of a vector space, polynomials form a crucial part. They help in understanding functions and transformations and underpin a significant portion of linear algebra and calculus. An insightful example of polynomials at work is their ability to be expressed as sums of simpler polynomials, leading us into the topic of a linear combination.

Linear Combination

A linear combination refers to an expression constructed from a set of terms. Specifically, it is created by multiplying each term by some constant and then adding the results. It is essential in understanding how various elements can come together to form something new.

In the setting of vector spaces, a linear combination of polynomials means that we can assemble a new polynomial by scaling and summing these polynomials. For example, if we have polynomials like:

• \[ f(x) = c_1 p_1(x) + c_2 p_2(x) + \ldots + c_k p_k(x) \]
• where \( c_1, c_2, \ldots, c_k \) are scalars, and \( p_1(x), p_2(x), \ldots, p_k(x) \) are polynomials.

The concept of linear combinations helps us explore possibilities in a vector space, like the one described in the problem where each polynomial in the vector space can be expressed using a linear combination of others. It exemplifies the flexibility and power inherent in vector spaces, where new mathematical structures are created from known elements.

Basis of a Vector Space

A basis of a vector space is a set of vectors that are linearly independent and span the entire space. This concept is crucial because it allows us to represent every element in the space as a linear combination of the basis elements. Understanding the basis helps us grasp the structure and dimensionality of vector spaces.

Characteristics of a basis:

• The elements of the basis must be linearly independent, meaning no element in the basis can be written as a linear combination of the others.
• Every vector in the space can be expressed as a unique combination of the basis elements, highlighting the completeness of the basis.

In polynomial vector spaces, a basis often consists of polynomial functions, such as \( p_1(x), p_2(x), \ldots, p_k(x) \). Given these polynomials, any other polynomial in that vector space can be constructed.

As shown in the original problem, verifying that at least one polynomial in the basis evaluates to a non-zero value at a certain point (like \( a \)) confirms the non-trivial nature of the polynomial vector space. This ensures that the basis genuinely spans the space since if all basis elements evaluated to zero at a certain point, it could contradict the span or lead to conceptual conflicts as demonstrated by forming zero polynomial when evaluated.