- #1
dmitriylm
- 39
- 2
Homework Statement
Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not?
*The digits should be in subscript.
How would I go about answering this?
dmitriylm said:Homework Statement
Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not?
*The digits should be in subscript.
How would I go about answering this?
Mark44 said:Let's call your set as described above S. There are three things you need to verify to say that S is a subspace of P3:
- The zero polynomial is in S.
- If p1 and p2 are any two polynomials in S, then p1 + p2 is in S.
- If c is any real constant and p1 is in S, the cp1 is also in S.
dmitriylm said:Homework Statement
Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not?
*The digits should be in subscript.
How would I go about answering this?
A subspace is a subset of a vector space that contains vectors that can be added together and multiplied by scalars, while still remaining within the original vector space.
P3 is the vector space of polynomials with degree less than or equal to 3, where the coefficients are real numbers. It is denoted as P3 = {a0 + a1x + a2x^2 + a3x^3 | a0, a1, a2, a3 ∈ ℝ}.
To determine if a set is a subspace of P3, you must check if it satisfies three conditions: closure under vector addition, closure under scalar multiplication, and contains the zero vector. If all three conditions are met, then the set is a subspace of P3.
The set {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} will be a subspace of P3 if and only if the zero polynomial, which is the polynomial with all coefficients equal to 0, is included in the set. This ensures that the set contains the zero vector and satisfies the closure under scalar multiplication condition.
One example of a polynomial that is not a subspace of P3 is 2 + 3x + 4x^2 + 5x^3. This polynomial does not satisfy the condition a0a3 - a1a2 = 0, as 2*5 - 3*4 ≠ 0. Therefore, it is not a subspace of P3.