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Find the rank and nullity of the linear transformation T(f(t) = f'(t) + 4f(t) from P₂ to P₂.

Find the rank and nullity of the linear transformation T(f(t) = f'(t) + 4f(t) from P₂ to P₂.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the rank and nullity of the linear transformation T(f(t)) = f''(t) + 4f(t) from P2 to P2.

**Problem:** Find the rank and nullity of the linear transformation \( T(f(t)) = f''(t) + 4f(t) \) from \( P_2 \) to \( P_2 \). **Explanation:** The given problem involves a linear transformation \( T \) where \( f(t) \) is a polynomial in the set \( P_2 \), which consists of all polynomials of degree at most 2. The transformation \( T \) maps a polynomial \( f(t) \) to the sum of its second derivative \( f''(t) \) and four times the original polynomial \( 4f(t) \). To find the rank and nullity, you will need to determine the null space (kernel) and the range of the transformation \( T \). The nullity is the dimension of the null space, and the rank is the dimension of the range. Since \( f(t) \in P_2 \), let \( f(t) = at^2 + bt + c \). Calculate \( f''(t) \) and substitute back into the equation to find polynomials in \( P_2 \) that satisfy \( T(f(t)) = 0 \). This problem involves understanding differential operators, polynomial spaces, and fundamental concepts of linear algebra such as basis, dimension, rank, and null space.
Transcribed Image Text:**Problem:** Find the rank and nullity of the linear transformation \( T(f(t)) = f''(t) + 4f(t) \) from \( P_2 \) to \( P_2 \). **Explanation:** The given problem involves a linear transformation \( T \) where \( f(t) \) is a polynomial in the set \( P_2 \), which consists of all polynomials of degree at most 2. The transformation \( T \) maps a polynomial \( f(t) \) to the sum of its second derivative \( f''(t) \) and four times the original polynomial \( 4f(t) \). To find the rank and nullity, you will need to determine the null space (kernel) and the range of the transformation \( T \). The nullity is the dimension of the null space, and the rank is the dimension of the range. Since \( f(t) \in P_2 \), let \( f(t) = at^2 + bt + c \). Calculate \( f''(t) \) and substitute back into the equation to find polynomials in \( P_2 \) that satisfy \( T(f(t)) = 0 \). This problem involves understanding differential operators, polynomial spaces, and fundamental concepts of linear algebra such as basis, dimension, rank, and null space.
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