# Proving Linear Independence of Polynomials with Non-Zero Degrees | Math Solution

• kash25
In summary, the statement that {p, q, pq} is linearly independent if and only if deg(p)>=1 and deg(q)>=1 is not correct. A counterexample can be found using deg(p) = -1 and deg(q) = -2, where p(x) = 1/x and q(x) = 1/x^2. However, this counterexample is incorrect as it does not follow the definition of polynomials, which are linear combinations of x^n where n>=0. The statement is actually true, and can be proven.
kash25
Linear independence!?

## Homework Statement

Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg(p)>=1 and deg(q)>=1.

## The Attempt at a Solution

I am pretty sure the statement to prove is incorrect.
If we use deg(p) = -1 and deg(q) = -2, we can easily show that the two are linearly independent (consider the functions p(x) = 1/x and q(x) = 1/x^2).
We can have k/x + l/x^2 = 0
then kx + l = 0.
Then we can differentiate and get k = 0 and l = 0, which disproves the statement.
How does this make any sense?

If you are talking about polynomials you are only talking about linear combinations of x^n where n>=0. There is so much more wrong with your counterexample, I don't know where to start... The statement is true, now try and prove it.

## What is linear independence of polynomials?

Linear independence of polynomials refers to a set of polynomials that cannot be expressed as a linear combination of each other. In other words, none of the polynomials in the set can be written as a multiple or sum of the other polynomials in the set.

## How can I prove the linear independence of polynomials?

To prove the linear independence of polynomials, you can use the method of setting up a linear combination and showing that the only solution is when all the coefficients are equal to zero. This can be done by setting up a system of equations and solving for the coefficients.

## What are non-zero degrees in polynomials?

Non-zero degrees in polynomials refer to the terms with coefficients that are not equal to zero. For example, in the polynomial 3x^2 + 5x + 1, the terms with coefficients 3, 5, and 1 are non-zero degrees.

## Can polynomials with non-zero degrees be linearly dependent?

Yes, polynomials with non-zero degrees can be linearly dependent if one or more polynomials in the set can be expressed as a linear combination of the other polynomials in the set. In other words, if one or more terms with non-zero coefficients can be eliminated by combining them with other terms, then the polynomials are linearly dependent.

## What is the importance of proving linear independence of polynomials?

Proving linear independence of polynomials is important in various mathematical and scientific applications. It allows us to determine whether a set of polynomials can be used as a basis for a vector space, to find the dimension of a vector space, and to solve systems of linear equations. It also has applications in fields such as engineering, physics, and computer science.

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