Can p₁ be written as a Linear Combination of p₂ and p₃?
What is a Linear Combination?
A vector (or polynomial) p₁ is a linear combination of p₂ and p₃ if there exist scalars a, b ∈ ℝ such that:
In the polynomial space P₃(ℝ), polynomials behave like vectors: we can add them and scale them. Checking for a linear combination reduces to asking whether a certain system of linear equations has a consistent solution.
Given Polynomials
Verify whether the first polynomial can be expressed as a linear combination of the other two in P₃(ℝ):
Step-by-Step Verification
Set Up the Equation
We want scalars a, b ∈ ℝ such that:
Substituting each polynomial:
Expand & Collect Like Terms
Compare Coefficients
Since two polynomials are equal if and only if all corresponding coefficients match, we extract one equation per degree:
| (1) | a + b = 1 | coeff. of x³ |
| (2) | −2a = −8 | coeff. of x² |
| (3) | 3a − 2b = 4 | coeff. of x |
| (4) | −a + 3b = 0 | constant term |
Solve Equations (1) & (2)
From equation (2):
Substituting a = 4 into equation (1):
Check Equations (3) & (4)
We must verify that a = 4, b = −3 satisfies the remaining equations.
Both remaining equations are violated. The system is inconsistent — no pair (a, b) can satisfy all four equations simultaneously.
No — p₁(x) = x³ − 8x² + 4x cannot be expressed as a linear combination of p₂ and p₃ in P₃(ℝ), because the resulting system of equations is inconsistent.
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