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Consider L: R2 R2 given by → 2x1 L + (+₁) = (₁²+1₂). X2 x2, Consider the two bases S = [e₁,e2] and F = [u₁, u2], where - (1¹). U₁ = and its inverse is (1), We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(er)]s_ [L(e2)]s) = (ii). 1 (b) The representation matrix B of L with respect to F is 2 B = ([L(ur)]F_ [L(u2)]x) = (1 7¹). 0 (c) The transition matrix T from F to S is U2 = T = ([1₁]s (u₂]s) = (11¹), T-1 = We can verify directly that 1 2 (11) 1 1 2 B=T-¹ AT = (¹₁¹) (²) (1)-67 - = 2 1 1

Consider L: R2 R2 given by → 2x1 L + (+₁) = (₁²+1₂). X2 x2, Consider the two bases S = [e₁,e2] and F = [u₁, u2], where - (1¹). U₁ = and its inverse is (1), We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(er)]s_ [L(e2)]s) = (ii). 1 (b) The representation matrix B of L with respect to F is 2 B = ([L(ur)]F_ [L(u2)]x) = (1 7¹). 0 (c) The transition matrix T from F to S is U2 = T = ([1₁]s (u₂]s) = (11¹), T-1 = We can verify directly that 1 2 (11) 1 1 2 B=T-¹ AT = (¹₁¹) (²) (1)-67 - = 2 1 1

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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(4) Consider L: R2 R2 given by Consider the two bases S = [e₁, e2] and F = [u₁, u2], where (¹). (1¹). x1 2x1 L - (^₂) = (2,2²4 ¹₂). x2 + U₁ = We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(en)]s_[L(e2)]s) = (ii) 9). 1 (b) The representation matrix B of L with respect to F is and its inverse is 2 B = [L(U₂)]F) · ([L(ur)]F_ [L(us)]x) = (1 7¹). 0 U2 = (c) The transition matrix T from F to S is T = ([m]s [1₂]s) = (11¹), B = T-¹ AT = T-1 We can verify directly that 1/ = 1 (41) 2 2 2 - 1 (1 1) (² 9) ( 1 ) = (²1¹). 2 1 0
Transcribed Image Text:(4) Consider L: R2 R2 given by Consider the two bases S = [e₁, e2] and F = [u₁, u2], where (¹). (1¹). x1 2x1 L - (^₂) = (2,2²4 ¹₂). x2 + U₁ = We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(en)]s_[L(e2)]s) = (ii) 9). 1 (b) The representation matrix B of L with respect to F is and its inverse is 2 B = [L(U₂)]F) · ([L(ur)]F_ [L(us)]x) = (1 7¹). 0 U2 = (c) The transition matrix T from F to S is T = ([m]s [1₂]s) = (11¹), B = T-¹ AT = T-1 We can verify directly that 1/ = 1 (41) 2 2 2 - 1 (1 1) (² 9) ( 1 ) = (²1¹). 2 1 0
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