Understanding the Linear Independence of Polynomials

Polynomials are fundamental in many areas of mathematics and are often used in various applications, including computer graphics, physics, and engineering. A key property of polynomials is linear independence, which is crucial for a variety of theoretical and practical purposes. In this article, we delve into why polynomials of different degrees are linearly independent and explore the concept through detailed mathematical reasoning.

Definition of Linear Independence

A set of functions or vectors {f_1(x), f_2(x), ..., f_n(x)} is considered linearly independent if the only solution to the equation c_1 f_1(x) c_2 f_2(x) ... c_n f_n(x) 0 is c_1 c_2 ... c_n 0, where c_i are constants. This definition is the cornerstone of understanding the linear independence of polynomials.

Polynomials of Different Degrees

Consider a set of polynomials {p_0(x), p_1(x), ..., p_n(x)}, where each p_i(x) is a polynomial of degree i. For example:

p_0(x) 1, degree 0 p_1(x) x, degree 1 p_2(x) x^2, degree 2 ... p_n(x) x^n, degree n

Argument for Linear Independence

Degree Consideration

If the linear combination c_0 p_0(x) c_1 p_1(x) ... c_n p_n(x) 0 holds for all x, the left-hand side is a polynomial of degree at most n, the highest degree among the polynomials.

Identifying Coefficients

Since this polynomial is equal to the zero polynomial, which has no terms, all coefficients must be zero. Specifically, the coefficient of x^k where k 0, 1, ..., n must equal zero.

Conclusion

Therefore, we must have c_0 c_1 ... c_n 0. This shows that no polynomial in the set can be expressed as a linear combination of the others, confirming their linear independence.

Summary

Polynomials of different degrees are linearly independent because any non-trivial linear combination of them can only equal the zero polynomial if all coefficients are zero. This property holds true for polynomials in general as long as they have distinct degrees.

Polynomials In General

Not all polynomials are linearly independent. They are independent when and only when you set up an equation like this:

a1 P1(x) - a2 P2(x) - ... - an Pn(x) 0

and find that the only way you can make this equation true is by setting all the a's equal to 0. For example, to show that 1, x, x^2 are linearly independent, you need to set:

a0 a1 x a2 x^2 0

at all times, independent of any particular x. The only way this can happen is by setting a0 a1 a2 0. Hence, 1, x, x^2 are all linearly independent.