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

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.
  • #1
kash25
12
0
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?
 
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  • #2


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.
 

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

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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