Math Problem Statement
Solution
The question is asking whether the vectors , , and from the space (polynomials of degree at most 2) are linearly independent.
Step-by-step solution:
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Definition: A set of vectors (or polynomials) is linearly independent if the only solution to the equation: is .
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Form the equation: Let's expand the linear combination: Expanding each term: Putting them all together: Grouping terms by powers of :
- Constant term:
- Linear term ():
- Quadratic term ():
This gives the system of equations:
c_1 &= 0 \quad \text{(from the constant term)} \\ -2c_1 + c_3 &= 0 \quad \text{(from the linear term)} \\ c_2 + c_3 &= 0 \quad \text{(from the quadratic term)} \end{aligned}$$ -
Solve the system:
- From , substitute into the second equation:
- Substitute into the third equation:
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Conclusion: The only solution to this system is , , and . Therefore, the vectors are linearly independent.
Correct Answer:
B. They are linearly independent.
Would you like more details or have any questions?
Here are five related questions you could explore:
- What is the difference between linear independence and linear dependence?
- Can vectors in higher-dimensional polynomial spaces be linearly dependent?
- How does the method for determining linear independence change with more vectors?
- What is the role of the zero vector in determining linear independence?
- How does linear independence relate to the concept of a basis?
Tip: To test linear independence, always set up the linear combination and analyze the resulting system of equations.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Linear Independence
Vector Spaces
Polynomial Spaces
Formulas
Linear combination: c1(1 - 2x) + c2(x^2) + c3(x + x^2) = 0
Theorems
Theorem of Linear Independence in Vector Spaces
Suitable Grade Level
University Level (Undergraduate)
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