Math Problem Statement
Find a basis p(x), q(x) for the kernel of the linear transformation L : P3(x) -> R defined by L(f(x)) = f'(-6) - f(1) where P3(x) is the vector space of polynomials in x with degree less than 3
Solution
To find a basis for the kernel of the linear transformation , where , follow these steps:
Step 1: Define the kernel of
The kernel of is the set of all polynomials such that . That is:
Step 2: Represent in
A general polynomial in has the form: where .
Step 3: Compute and
The derivative of is: Thus: And:
Step 4: Solve
Substituting into , we have: Simplify: This simplifies further to:
Step 5: Write the kernel in parametric form
Let , where . Then: a_0 = -13t, \quad a_1 = s \quad \text{(let a_1 = s, free variable)}. Thus: Factor and to express in terms of basis elements:
Step 6: Extract the basis
The kernel is spanned by:
Final Answer:
The basis for the kernel is:
Would you like a deeper explanation or additional examples?
- What is the geometric interpretation of the kernel of a linear transformation?
- How does the kernel relate to the rank-nullity theorem?
- Could this kernel basis change if the degree of were higher?
- How would the calculation change if for general and ?
- What is the relationship between the kernel of and the space ?
Tip: To verify your basis, plug it back into and confirm that the transformation evaluates to zero for each basis element.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Kernels of Linear Transformations
Polynomial Functions
Formulas
f(x) = a_0 + a_1x + a_2x^2
f'(x) = a_1 + 2a_2x
f'(-6) = a_1 - 12a_2
f(1) = a_0 + a_1 + a_2
L(f(x)) = f'(-6) - f(1)
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate (Linear Algebra Course)
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