Math Problem Statement
Solution
To solve the problem, we need to find the coordinate vector of relative to the given basis .
Step-by-Step Solution
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Express as a linear combination of the basis vectors: Let: Substitute the basis vectors:
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Expand the right-hand side: Distribute and : Combine like terms:
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Equate coefficients: Compare coefficients of , , and on both sides:
- Constant term:
- Coefficient of :
- Coefficient of :
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Solve the system of equations:
- From , substitute into the other equations:
- Substitute and into :
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Coordinate vector: The coordinate vector of relative to the basis is:
Final Answer:
The coordinate vector is:
Would you like further clarification on any of these steps?
Here are 5 related questions to expand your understanding:
- How do you verify that the calculated coordinate vector correctly reconstructs ?
- What properties of a basis make such a decomposition possible?
- How would the solution change if the basis vectors included higher powers of ?
- Why is the system of equations derived by equating coefficients always consistent?
- How does this concept relate to solving systems of linear equations in matrix form?
Tip: Always confirm that the given set of vectors forms a basis before calculating coordinate vectors.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Coordinate Vectors
Polynomial Spaces
Formulas
p(x) = c1 * p1 + c2 * p2 + c3 * p3
Equating coefficients of like terms: c1 + c2 + c3 = constant, etc.
Theorems
Linear Independence
Basis and Dimension of Vector Spaces
Suitable Grade Level
Grades 11-12 or early undergraduate
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