Math Problem Statement
Suppose p,q \in P(\mathbb{C}) are nonconstant polynomials with no zeros in common. Let r \in P_{n-1}(\mathbb{C}) and s \in P_{m-1}(\mathbb{C}). Let m = deg(p) and n = deg(q). Define T : P_{n-1}(\mathbb{C}) \times P_{m-1}(\mathbb{C}) \rightarrow P_{m+n-1}(\mathbb{C}) by T(r,s) = rp + sq. Show that the linear map T is injective and surjective.
Solution
To show that the linear map is injective and surjective, we analyze it step by step.
Definitions and Setup
Let:
- denote the space of polynomials of degree at most with coefficients in .
- and are polynomials of degrees and , respectively, with no common zeros.
- defined by , where and .
Step 1: Show that is injective
Approach:
To show injectivity, we need to prove that if , then and .
- Suppose .
- Rearrange to get .
- Since and have no common zeros, the polynomial can only vanish if both and .
Reasoning:
- The lack of common zeros ensures that and are linearly independent in the space of polynomials.
- Therefore, and must both vanish for the equation to hold.
Thus, is injective.
Step 2: Show that is surjective
Approach:
To show surjectivity, we need to prove that for any polynomial , there exist and such that .
- Write . This means has degree at most .
- Since and are linearly independent and span , any can be written as for some and .
Constructing and :
- Use the Euclidean algorithm to divide by , yielding , where and .
- Since is independent of , can also be expressed in terms of .
Thus, is surjective.
Conclusion
Since is both injective and surjective, it is a linear isomorphism. This completes the proof.
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Related Questions
- What are the implications of and having no common zeros?
- How does the degree of influence the surjectivity of ?
- Can the proof be extended to real polynomials, , instead of ?
- How does linear independence of and affect the space they span?
- What happens if and have common zeros?
Tip
To verify injectivity or surjectivity of a linear map, always test the kernel (for injectivity) and image (for surjectivity) explicitly.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Polynomial Functions
Injectivity and Surjectivity of Linear Maps
Vector Spaces
Formulas
T(r, s) = rp + sq
Degree of a polynomial: deg(rp + sq) ≤ max(deg(rp), deg(sq))
Theorems
Linear Independence of Polynomials
Properties of Polynomial Spaces over Complex Numbers
Suitable Grade Level
Undergraduate Mathematics
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