Math Problem Statement
Solution
Let's analyze and solve the given problems step by step.
Problem 1:
We have a -vector space of finite dimension and . We assume is nilpotent with index of nilpotence .
(a) Write the propositions and using quantifiers.
Solution:
- :
- :
(b) Show that for a certain , the family is linearly independent.
Solution: We assume is nilpotent with index , i.e., . Let us consider such that . We need to show that the family is linearly independent.
Assume there exist scalars such that: Applying repeatedly to this equation, we obtain:
\alpha_0 f(x) + \alpha_1 f^2(x) + \alpha_2 f^3(x) + \cdots + \alpha_{p-2} f^{p-1}(x) + \alpha_{p-1} f^p(x) = 0 \\ \alpha_0 f^2(x) + \alpha_1 f^3(x) + \cdots + \alpha_{p-2} f^p(x) = 0 \\ \vdots \\ \alpha_0 f^{p-1}(x) + \alpha_1 f^p(x) = 0 \\ \alpha_0 f^p(x) = 0 \end{cases}$$ Since $$f^p(x) = 0$$, the last equation is trivial. The second last equation $$\alpha_0 f^{p-1}(x) = 0$$ implies $$\alpha_0 = 0$$ because $$f^{p-1}(x) \neq 0$$. Repeating this argument, we conclude $$\alpha_1 = \alpha_2 = \cdots = \alpha_{p-1} = 0$$. Hence, the family $$(x, f(x), f^2(x), \ldots, f^{p-1}(x))$$ is linearly independent. **(c)** Deduce that $$f^n = 0_{\mathcal{L}(E)}$$. **Solution:** Since $$f$$ is nilpotent with index $$p$$ and $$p \leq n$$, we have: $$f^p = 0_{\mathcal{L}(E)}$$ Given that $$p \leq n$$, it follows that: $$f^n = 0_{\mathcal{L}(E)}$$ --- **Problem 2:** We suppose in this question that $$p = n$$. According to 1.b, $$E$$ therefore has a basis of the form: $$(x, f(x), f^2(x), \ldots, f^{n-1}(x))$$ **Solution:** Since $$p = n$$, we have shown in 1.b that the family $$(x, f(x), f^2(x), \ldots, f^{n-1}(x))$$ is linearly independent and spans $$E$$. Therefore, it forms a basis of $$E$$. --- Do you want details or have any questions about the solutions? Here are five related questions to further your understanding: 1. What is the definition of a nilpotent linear operator? 2. How do we determine the index of nilpotence for a given operator? 3. What are the properties of a vector space with a nilpotent operator? 4. Can you provide an example of a nilpotent operator in a 3-dimensional vector space? 5. How does the concept of nilpotent operators relate to the Jordan canonical form? **Tip:** When dealing with nilpotent operators, always remember to verify the highest power of the operator that results in the zero transformation to find the index of nilpotence.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Nilpotent linear operators
Linear algebra
Vector spaces
Linear independence
Formulas
fp = 0 L(E)
fp−1 ≠ 0 L(E)
Linear independence of (x, f(x), f^2(x), …, f^(p-1)(x))
Theorems
Nilpotent operator theorem
Basis of a vector space
Suitable Grade Level
University level
Related Recommendation
Nilpotence of Operator u in Finite-Dimensional Vector Space E
Dim Ker f for Nilpotent Operator f in 3-Dimensional Vector Space
Proof of Nilpotency: Linear Operator with Trace Conditions
Defining a Linear Map and Proving Dimension Conditions in Vector Spaces
Prove Real Eigenvalues of Hermitian Matrices and Nilpotency of T