Show that two polynomials cannot span P2.

In summary: Hi In summary, the conversation involves discussing the dimension of a space, specifically the space of polynomials of degree less than or equal to 2. It is mentioned that two polynomials cannot span this space as they only have a dimension of 2. The conversation then shifts to finding a basis for a given set of vectors, and there is some confusion about whether to use rows or columns when forming a matrix. However, it is clarified that it doesn't matter which form is used as it will lead to the same result.
  • #1
kingkong69
22
0
Hi
I know the dimension is 3, two polynomials has dimension 2 only so it cannot span P2.
How would I go about showing it if I were to write it down mathematically?
Thanks
 
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  • #2
kingkong69 said:
Hi
I know the dimension is 3, two polynomials has dimension 2 only so it cannot span P2.
How would I go about showing it if I were to write it down mathematically?
Thanks



Do you mean P2 is the space of all polynomials (over some field, say) of degree less than or equal to 2?

Well, take two such pol's [itex]ax^2+bx+c\,\,,\,\,a'x^2+b'x+c'\,\,[/itex] , with a,b,c,a',b',c' elements of the field, and show that

[itex]\alpha(ax^2+bx+c)+\beta(a'x^2+b'x+c')\,\,,\,\, \alpha \,,\,\beta \,\,[/itex] in the field, cannot possibly give all the elements of [itex]P_2[/itex]...

DonAntonio
 
  • #3
Hi
thanks i get u.

i have a question for you.
when I am given a set of vectors, i am asked to find a basis for it.

I solve it to Reduced row echolon form and get the basis.

But sometimes i have to write the vectors in columns, sometimes in rows for the matrice.
Sorry for being unclear

Do i need to check if the vectors are linearly independent in column space form? and then apply it in row space if they are dependent?

Thanks DonAntonio
 
  • #4
kingkong69 said:
Hi
thanks i get u.

i have a question for you.
when I am given a set of vectors, i am asked to find a basis for it.


*** I'm guessing you actually mean that you have to find a basis for the space generated by that given set of vectors...right? ***


I solve it to Reduced row echolon form and get the basis.


*** Once again, I'm guessing you mean you write every polynomial as a row or column vector (it never matters, believe me) and then you

form a matrix with those rows (vectors) and then you reduce it...right? ***



But sometimes i have to write the vectors in columns, sometimes in rows for the matrice.
Sorry for being unclear



It doesn't matter rows or columns: it'sjust the same at the end.

DonAntonio
 
  • #5
for your question. To show that two polynomials cannot span P2, we can use a proof by contradiction. We assume that two polynomials, say p1(x) and p2(x), can span P2. This means that any polynomial of degree 2 or less can be written as a linear combination of p1(x) and p2(x).

Let's consider a polynomial q(x) = ax^2 + bx + c, where a, b, and c are constants. Since q(x) is of degree 2, it can be written as a linear combination of p1(x) and p2(x), i.e. q(x) = kp1(x) + lp2(x), where k and l are constants.

Now, let's substitute x = 0 into both sides of the equation. We get q(0) = k*p1(0) + l*p2(0). Since p1(0) and p2(0) are constants, we can rewrite this as q(0) = m, where m is some constant.

But we know that q(0) = c, since we substituted x = 0. Therefore, we have c = m, which means that c is a constant. This is a contradiction, as c can be any constant, not just a specific one.

Since we have reached a contradiction, our initial assumption that two polynomials can span P2 must be false. Therefore, two polynomials cannot span P2. This shows that the dimension of P2 is indeed 3, and two polynomials cannot span it.

I hope this helps to clarify the concept. Let me know if you have any further questions.
 

1. What is P2?

P2 refers to the set of all polynomials of degree 2 or less.

2. How do you determine if two polynomials span P2?

To determine if two polynomials span P2, we need to see if every polynomial in P2 can be written as a linear combination of the two given polynomials.

3. Why can't two polynomials span P2?

Two polynomials cannot span P2 because P2 is a vector space of infinite dimension, while two polynomials can only form a subspace of dimension 2. This means that they cannot represent all the possible polynomials in P2.

4. Can more than two polynomials span P2?

Yes, more than two polynomials can span P2. In fact, a set of three or more linearly independent polynomials can span P2, as long as they satisfy the necessary conditions for spanning a vector space.

5. How can you prove that two polynomials cannot span P2?

To prove that two polynomials cannot span P2, we can show that there exists at least one polynomial in P2 that cannot be written as a linear combination of the two given polynomials. This would demonstrate that the two polynomials do not span P2.

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