How To Write The Polynomial P = 5t^2 - 5t + 7 As A Linear Combination Of P1 = T^2 - 2t + 1, P2 = T + 2, And P3 = 2t^2 - T In The Space P2 = {at^2 + Bt + C / A, B, C ∈ R}?
In this comprehensive exploration, we delve into the vector space P₂, which comprises polynomials of degree at most 2. Our focus centers on expressing a given polynomial p as a linear combination of other polynomials within this space. This is a fundamental concept in linear algebra, with applications spanning various fields, including computer graphics, data analysis, and physics. We will address two distinct instances of this problem, each requiring a unique approach to determine the appropriate coefficients for the linear combination. Polynomials play a crucial role in mathematical modeling and approximation, and understanding how to manipulate them within vector spaces like P₂ is essential for tackling complex problems. The vector space P₂ is defined as the set of all polynomials of the form at² + bt + c, where a, b, and c are real numbers. This space exhibits the properties of a vector space, allowing for addition and scalar multiplication of its elements (polynomials). Our primary task is to find scalars that, when multiplied by the given polynomials p₁, p₂, and p₃, and then summed, result in the target polynomial p. This process involves setting up a system of linear equations and solving for the unknown scalars. The linear combination of vectors is a cornerstone of linear algebra, enabling us to express vectors in terms of a set of basis vectors. In the context of polynomials, this means representing a polynomial as a sum of scalar multiples of other polynomials. The ability to do this is crucial for simplifying expressions, solving equations, and performing various transformations on polynomials.
Problem Statement
Consider the vector space P₂ = {at² + bt + c | a, b, c ∈ ℝ}, which represents the set of all polynomials of degree at most 2 with real coefficients. We are given three vectors (polynomials) within this space:
- p₁ = t² - 2t + 1
- p₂ = t + 2
- p₃ = 2t² - t
We aim to solve the following two problems:
a) Express the vector p = 5t² - 5t + 7 as a linear combination of p₁, p₂, and p₃.
b) (The second part of the problem is missing, but we will address the general method for expressing a vector as a linear combination.)
Part a: Expressing p as a Linear Combination of p₁, p₂, and p₃
To express p as a linear combination of p₁, p₂, and p₃, we need to find scalars c₁, c₂, and c₃ such that:
p = c₁ p₁ + c₂ p₂ + c₃ p₃
Substituting the given polynomials, we get:
5t² - 5t + 7 = c₁ (t² - 2t + 1) + c₂ (t + 2) + c₃ (2t² - t)
Expanding the right side, we have:
5t² - 5t + 7 = c₁ t² - 2c₁ t + c₁ + c₂ t + 2c₂ + 2c₃ t² - c₃ t
Now, we group the terms with the same powers of t:
5t² - 5t + 7 = (c₁ + 2c₃) t² + (-2c₁ + c₂ - c₃) t + (c₁ + 2c₂)
For this equation to hold for all values of t, the coefficients of the corresponding powers of t must be equal. This gives us a system of linear equations:
- c₁ + 2c₃ = 5
- -2c₁ + c₂ - c₃ = -5
- c₁ + 2c₂ = 7
We now solve this system of equations for c₁, c₂, and c₃. This system of linear equations represents the core of the problem. Solving it allows us to find the scalar coefficients that express the target polynomial as a combination of the given polynomials. The process of solving this system often involves techniques such as Gaussian elimination, matrix inversion, or substitution. The solution provides the unique set of scalars that satisfy the linear combination equation. The system of linear equations is derived from the polynomial equation by equating the coefficients of corresponding powers of t. Each equation in the system represents a constraint that the coefficients c₁, c₂, and c₃ must satisfy. Solving this system is a crucial step in determining the linear combination. The method used to solve the system can vary depending on the specific equations, but the goal remains the same: to find the values of the unknown coefficients. Understanding how to set up and solve these systems is fundamental to linear algebra and its applications.
Solving the System of Equations
From equation (3), we can express c₁ in terms of c₂:
c₁ = 7 - 2c₂
Substitute this into equation (1):
(7 - 2c₂) + 2c₃ = 5
2c₃ = 2c₂ - 2
c₃ = c₂ - 1
Now, substitute c₁ and c₃ into equation (2):
-2(7 - 2c₂) + c₂ - (c₂ - 1) = -5
-14 + 4c₂ + c₂ - c₂ + 1 = -5
4c₂ = 8
c₂ = 2
Now, we can find c₁ and c₃:
c₁ = 7 - 2(2) = 3
c₃ = 2 - 1 = 1
Thus, we have c₁ = 3, c₂ = 2, and c₃ = 1.
Therefore, the linear combination is:
p = 3p₁ + 2p₂ + p₃
We can verify this by substituting the values:
3(t² - 2t + 1) + 2(t + 2) + (2t² - t) = 3t² - 6t + 3 + 2t + 4 + 2t² - t = 5t² - 5t + 7
This confirms that our solution is correct. The solution to the system of equations provides the specific scalar coefficients that enable us to express the target polynomial as a linear combination of the given polynomials. These coefficients are unique and represent the precise contribution of each polynomial to the overall combination. Once the coefficients are found, we can verify the solution by substituting them back into the original linear combination equation. This verification step ensures the accuracy of our calculations and confirms that the linear combination indeed produces the desired polynomial. The process of finding these coefficients and verifying the solution is a fundamental skill in linear algebra, with applications in various mathematical and computational contexts.
Part b: General Method for Expressing a Vector as a Linear Combination
To express a vector (in this case, a polynomial) as a linear combination of other vectors, the general method involves setting up a system of linear equations. Let's say we have a vector p and we want to express it as a linear combination of vectors v₁, v₂, ..., vₙ:
p = c₁ v₁ + c₂ v₂ + ... + cₙ vₙ
- Write out the vectors explicitly: Substitute the expressions for p and the vᵢ vectors.
- Expand the linear combination: Distribute the scalars cᵢ and combine like terms.
- Equate coefficients: Equate the coefficients of the corresponding components (or powers of t, in the case of polynomials) on both sides of the equation.
- Form a system of linear equations: The equations obtained in step 3 form a system of linear equations with the cᵢ as unknowns.
- Solve the system of equations: Use any suitable method (e.g., Gaussian elimination, matrix inversion, substitution) to solve for the cᵢ.
- Write the linear combination: Substitute the values of cᵢ back into the linear combination equation.
This method can be applied to any vector space, not just P₂. The key is to understand how the vector space operations (addition and scalar multiplication) work and how to equate components or coefficients. The general method for expressing a vector as a linear combination provides a systematic approach to solving this type of problem. This method is applicable not only to polynomials but also to vectors in any vector space. The steps involved include writing out the vectors, expanding the linear combination, equating coefficients, forming a system of linear equations, solving the system, and finally writing the linear combination. This process demonstrates the power of linear algebra in representing and manipulating vectors in a consistent and organized manner. The ability to apply this method is a fundamental skill for anyone working with linear systems and vector spaces.
Conclusion
In this article, we successfully expressed a polynomial p as a linear combination of other polynomials in the vector space P₂. We demonstrated the process of setting up and solving a system of linear equations to find the appropriate coefficients. Furthermore, we outlined the general method for expressing a vector as a linear combination, which is applicable to various vector spaces. Understanding these concepts is crucial for working with polynomials and linear algebra in general. The ability to express a polynomial as a linear combination is a powerful tool in mathematics and its applications. It allows us to represent polynomials in different forms, which can be useful for simplifying expressions, solving equations, and performing various transformations. The process involves understanding the structure of vector spaces, setting up and solving systems of linear equations, and interpreting the results in the context of the problem. This skill is essential for anyone working with polynomials in advanced mathematical settings. The concepts and techniques discussed in this article provide a solid foundation for further exploration of linear algebra and its applications.