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= (a) Find a linear transformation L : F² → F² so that L ‡ 0 and L²= 0. (Hint: Let L(e₁) e2 and L(e₂) = 0.). Find a linear transformation L : F³ → F³ so that L, L² ‡ 0, but L³ 0. (Hint: shift the standard basis.) Find a linear transformation L: Fn → Fn so that L, L², . Ln-¹ ‡ 0, but L” = 0. "..." = - (b) Find a 2 × 2 matrix A so that A ‡ 0 and A² 0. Find a 3 × 3 matrix A so that A, A² ‡ 0 and A³ = 0. Find an n × n matrix A so that A, A², ... An-1 0 and An = 0. (Hint: Use the matrices of the linear transformations from (a).) . =

= (a) Find a linear transformation L : F² → F² so that L ‡ 0 and L²= 0. (Hint: Let L(e₁) e2 and L(e₂) = 0.). Find a linear transformation L : F³ → F³ so that L, L² ‡ 0, but L³ 0. (Hint: shift the standard basis.) Find a linear transformation L: Fn → Fn so that L, L², . Ln-¹ ‡ 0, but L” = 0. "..." = - (b) Find a 2 × 2 matrix A so that A ‡ 0 and A² 0. Find a 3 × 3 matrix A so that A, A² ‡ 0 and A³ = 0. Find an n × n matrix A so that A, A², ... An-1 0 and An = 0. (Hint: Use the matrices of the linear transformations from (a).) . =

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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(a) Find a linear transformation \( L : F^2 \to F^2 \) so that \( L \neq 0 \) and \( L^2 = 0 \). (Hint: Let \( L(e_1) = e_2 \) and \( L(e_2) = 0 \).) Find a linear transformation \( L : F^3 \to F^3 \) so that \( L, L^2 \neq 0 \), but \( L^3 = 0 \). (Hint: shift the standard basis.) Find a linear transformation \( L : F^n \to F^n \) so that \( L, L^2, \ldots, L^{n-1} \neq 0 \), but \( L^n = 0 \). (b) Find a \( 2 \times 2 \) matrix \( A \) so that \( A \neq 0 \) and \( A^2 = 0 \). Find a \( 3 \times 3 \) matrix \( A \) so that \( A, A^2 \neq 0 \) and \( A^3 = 0 \). Find an \( n \times n \) matrix \( A \) so that \( A, A^2, \ldots, A^{n-1} \neq 0 \) and \( A^n = 0 \). (Hint: Use the matrices of the linear transformations from (a).)
Transcribed Image Text:(a) Find a linear transformation \( L : F^2 \to F^2 \) so that \( L \neq 0 \) and \( L^2 = 0 \). (Hint: Let \( L(e_1) = e_2 \) and \( L(e_2) = 0 \).) Find a linear transformation \( L : F^3 \to F^3 \) so that \( L, L^2 \neq 0 \), but \( L^3 = 0 \). (Hint: shift the standard basis.) Find a linear transformation \( L : F^n \to F^n \) so that \( L, L^2, \ldots, L^{n-1} \neq 0 \), but \( L^n = 0 \). (b) Find a \( 2 \times 2 \) matrix \( A \) so that \( A \neq 0 \) and \( A^2 = 0 \). Find a \( 3 \times 3 \) matrix \( A \) so that \( A, A^2 \neq 0 \) and \( A^3 = 0 \). Find an \( n \times n \) matrix \( A \) so that \( A, A^2, \ldots, A^{n-1} \neq 0 \) and \( A^n = 0 \). (Hint: Use the matrices of the linear transformations from (a).)
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